












y 

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| - ; - 

Robert Beall, 
Bookseller and Stationer, 
495 Pennsylvania Ave., 



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F I E L D-B O 0 K 

FOR 


RAILROAD ENGINEERS. 

1 4 

CONTAINING 


FORMULA 

* % 

FOR LAYING OUT CURVES, DETERMINING FROG ANGLES, LEVELLING, 
CALCULATING EARTH-WORK, ETC., ETC., 

TOGETHER WITH 


TABLES 


OF RADII, ORDINATES, DEFLECTIONS, LONG CHORDS, MAGNETIC VARIA' 
TION, LOGARITHMS, LOGARITHMIC AND NATURAL SINES, 

TANGENTS, ETC., ETC. 

V v -1 

'.by * 

i n/VOr^^W 

JOHN II. II K X (• K. A.SI., 

CIVIL ENGINEER. 



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5 

1 

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3 

f 0 


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.1 


3 


NEW YORK: 

D. APPLETON & COMPANY, 549 & 551 BROADWAY. 
LONDON: 1G LITTLE BRITAIN 
1873. 



i 


V 




Entered, according to Act of Congress, in the year 1854, 
By D. APPLETON & CO., 

In the Clerk’s Office of the District Court of the United States 
for the Southern District of New York. 













PREFACE. 


The object of the present work is to supply a want very 
generally felt by Assistant Engineers on Railroads. Books 
of convenient form for use in the field, containing the ordi 
nary logarithmic tables, are common enough; but a book 
combining with these tables others peculiar to railroad 
work, and especially the necessary formula for laying out 
curves, turnouts, crossings, &c., is yet a desideratum. 
These formula, after long disuse perhaps, the engineer is 
often called upon to apply at a moment’s notice in the 
field, and he is, therefore, obliged to carry with him in 
manuscript such methods as he has been able to invent or 
collect, or resort to what has received the veiy appropriate 
name of “ fudging.” This the intelligent engineer always 
considers a reproach; and he will, therefore, it is hoped, 
receive with favor any attempt to make a resort to it inex¬ 
cusable. 

Besides supplying the want just alluded to, it was thought 
that some improvements upon former methods might be 
made, and some entirely new methods introduced. Among 
the processes believed to be original may be specified 
those in §§41-48, on Compound Curves, in Chapter II., 
on Parabolic Curves, in §§ 106 - 109, on Vertical Curves, 
and in the article on Excavation and Embankment. It is 



VI 


PREFACE. 


but just to add, that a great part of what is said on Reversed 
Curves, Turnouts, and Crossings, and most of the Miscel¬ 
laneous Problems, are the result of original investigations. 
In the remaining portions, also, many simplifications have 
been made. In all parts the object has been to reduce the 
operation necessary in .the field to a single process, indi¬ 
cated by a formula standing on a line by itself, and distin¬ 
guished by a GP. This could not be done in all cases, as 
will be readily seen ori examination. Certain preliminary 
steps were sometimes necessary, and these, whenever it 
was practicable, have been indicated by ivords in italics. 

Of the methods given for Compound Curves, that in 
§ 46 will be found particularly useful, from the great variety 
of applications of which it is susceptible. 

Methods of laying out Parabolic Curves are here given, 
that those so disposed may test their reputed advantages. 
Two things are certainly in their favor; they are adapted 
to unequal as well as equal tangents, and their curvature 
generally decreases towards both extremities, thus making 
the transition to and from a straight line easier. Some 
labor has been given to devising convenient ways of laying 
out these curves. The method of determining the radius 
of curvature at certain points is believed to be entirely 
new. Better processes, however, may already exist, par¬ 
ticularly in France, where these curves are said to be in 
genera] use. 

The mode of calculating Excavation and Embankment 
here presented, will, it is thought, be found at least as sim¬ 
ple and expeditious as those commonly used, with the ad¬ 
vantage over most of them in point of accuracy. The usual 
Tables of Excavation and Embankment have been- omitted. 
To include all the varieties of slope, width of road-bed, and 
depth of cutting, they must be of great extent, and unfitted 


PREFACE. 


rii 

for a field-book. Even then they apply only to ground 
whose cross-section is level, though often used in a manner 
shown to be erroneous in § 128. When the cross-section 
of the ground is level, the place of the tables is supplied by 
the formula of § 119, and when several sections are calcu¬ 
lated together, as is usually the case, and the work is ar¬ 
ranged in tabular form, as in § 120, the calculation is be¬ 
lieved to be at least as short as by the most extended tables. 
The correction in excavation on curves (§ 129) is not 
known to have been introduced elsewhere. 

In a work of this kind, brevity is an essential feature. 
The form of “Problem” and “Solution” has, therefore, 
been adopted, as presenting most concisely the thing to be 
done and the manner of doing it. Every solution, how¬ 
ever, carries with it a demonstration, which is deemed an 
equally essential feature. These demonstrations, with a 
few unavoidable exceptions, principally in Chapter II., pre¬ 
suppose a knowledge of nothing beyond Algebra, Geome¬ 
try, and Trigonometry. The result is in general expressed 
by an algebraic formula, and not in words. Those familiar 
with algebraic symbols need not be told how much more 
intelligible and quickly apprehended a process becomes 
when thus expressed. Those not familiar with these sym¬ 
bols should lose no time in acquiring the ready use of a 
language so direct and expressive. It may be remarked 
that it was no part of the author’s design to furnish a col¬ 
lection of mere “ rules,” professing to require only an abil¬ 
ity to read for their successful application. Rules can sel¬ 
dom be safely applied without a thorough understanding of 
the principles on which they rest, and such an understand¬ 
ing, in the present case, implies a knowledge of algebraic 
formulae. 

The tables here presented will, it is hoped, prove relia 


PREFACE. 


viii 

ble. Those specially prepared for this work have been 
computed with great care. The values have in some cases 
been carried out farther than ordinary practice requires, in 
order that interpolated values may be obtained from them 
more accurately. For the greater part of the material 
composing the Table of Magnetic Variation the author is 
indebted to Professor Bache, whose distinguished ability ir 
conducting the operations of the Coast Survey is equalled 
only by his desire to diffuse its results. The remaining 
tables have been carefully examined by comparing them 
with others of approved reputation for accuracy. Many 
errors have in this way been detected in some of the tables 
of corresponding extent in general use, particularly in the 
Table of Squares, Cubes, &c., and the Tables of Logarith¬ 
mic and Natural Sines, Cosines, &c. The number of tables 
might have been greatly increased, but for an unwillingness 
to insert any thing not falling strictly within the plan of the 
work or not resting on sufficient authority. 

J. B. II. 


Boston, February , 1854. 


TABLE OF CONTENTS. 


CHAPTER I. 

CIRCULAR CURVES. 

Article I. — Simple Curves. 

SECT. TAGS 

2. Definitions. Propositions relating to the circle . . 1 

4. Angle of intersection and radius given, to find the tangent 3 

5. Angle of intersection and tangent given, to find the radius 3 

6. Degree of a curve. 4 

7. Deflection angle of a curve.4 

A. Method hy Deflection Angles. 

9. Radius given, to find the deflection angle .... 4 

10. Deflection angle given, to find the radius ... 1 

11. Angle of intersection and tangent given, to find the deflection 

angle . . .5 

12. Angle of intersection and deflection angle given, to find the 

tangent ..5 

13. Angle of intersection and deflection angle given, to find the 

length of the curve.6 

14. Deflection angle given, to lay out a curve .... 7 

16. To find a tangent at any station.8 

B. Method hy Tangent and Chord Deflections. 

17. Definitions ... .... .8 

18. Radius given, to find the tangent deflection and chord deflection 9 

19. Deflection angle given, to find the chord deflection . . 9 

21. To find a tangent at any station.9 

22. Chord deflection given, to lay out a cnrv# . . . . 10 





K 


TABLE OF CONTENTS. 


C. Ordinates. 

SECT. PA«3* 

24. Definition.11 

25. Deflection angle or radius given, to find ordinates . 11 

26. Approximate value for middle ordinate . . . . 13 

27. Method of finding intermediate points on a curve approxi¬ 

mately . . . . . 14 

D. Curving Rails. 

29. Deflection angle or radius given, to find the ordinate for curv¬ 

ing rails . .14 

Article II. — Reversed and Compound Curves. 

30. Definitions . ..15 

31. Radii or deflection angles given, to lay out a reversed or com¬ 

pound curve . ... .16 

A. Reversed Curves. 

32. Reversing point when the tangents are parallel . . 16 

33. To find the common radius when the tangents are parallel 16 

34. One radius given, to find the other when the tangents are par¬ 

allel .... .17 

35. Chords given, to find the radii when the tangents are parallel 18 

36. Radii given, to find the chords when the tangents are parallel 18 

37. Common radius given, to run the curve when the tangents are 

not parallel.19 

38. One radius given, to find the other when the tangents are not 

parallel.19 

39. To find the common radius w£en the tangents are not parallel 21 

40. Second method of finding the common radius when the tan¬ 

gents are not parallel. 22 

B. Compound Curves. 

41. Common tangent point .... .23 

42. To find a limit in one direction of each radius . . 24 

44. One radius given, to find the other.25 

45. Second method of finding one radius when the other is given 26 

46. To find the two radii.27 

47. To find the tangents of the two branches . . . . 29 

48. Second method of finding the tangents of the two branches . 30 







TABLE OF CONTENTS. 


SI 


Article III. — Turnouts and Crossings. 

HECT. PAGE 

49. Definitions.. . . . 3i 

A. Turnout from Straight Lines. 

50. Radius given, to find the frog angle and the position of the frog 32 

51. Frog angle given, to find the radius and the position of the frog 33 

52. To find mechanically the proper position of a given frog . 34 

53. Turnouts that reverse and become parallel to the main track 34 

54. To find the second radius of a turnout reversing opposite the 

frog. ... 35 

B. Crossings on Straight Lines. 

55. References to proper problems..36 

56. Radii given, to find the distance between switches . 36 

C. Turnout from Curves. 

57. Frog angle given, to find the radius and the position of the frog 38 

58. To find mechanically the proper position of a given frog . 41 

59 Proper angle for frogs that they may come at the end of a rail 41 
60. Radius given, to find the frog angle and the position of the frog 42 

62. Turnout to reverse and become parallel to the main track . 44 

D. Crossings on Curves. 

63. References to proper problems.45 

64. Common radius given, to find the central angles and chords 45 

Article TV . — Miscellaneous Problems. 

65. To find the radius of a curve to pass through a given point 46 

66. To find the tangent point of a curve to pass through a given 

point.47 

67. To find, the distance to the curve from any point on the tan¬ 

gent.47 

68. Second method for passing a curve through a given point . 47 

69. To find the proper chord for any angle of deflection . . 48 

70. To find the radius when the distance from the intersection 

point to the curve is given.48 

71 To find the distance from the intersection point to the curve 

when the radius is given ... ... 49 




xn 


TABLE OF CONTENTS. 


8ECT. PAG1 

72. To find the tangent point of a curve that shall pass through a 

given point ........ 50 

7.3. To find the radius of a curve without measuring angles . 51 

74. To find the tangent points of a curve without measuring an¬ 

gles .... . ... 59 

75. To find the angle of intersection and the tangent points when 

the point of intersection is inaccessible . . . . 52 

76. To lay out a curve when obstructions occur . . 55 

77. To change the tangent point of a curve, so that it may pass 

through a given point.56 

78. To change the radius of a curve, so that it may terminate in 

a tangent parallel to its present tangent . . . .57 

79. To find the radius of a curve on a track already laid . . 58 

80. To draw a tangent to a given curve from a given point . . 59 

81. To flatten the extremities of a sharp curve . . . . 59 

82. To locate a curve without setting the instrument at the tan¬ 

gent point ..60 

83. To measure the distance across a river . . . 63 

CHAPTER II. 

PARABOLIC CURVES. 

Article I. — Locating Parabolic Curves. 


84. Propositions relating to the parabola. 65 

85. To lay out a parabola by tangent deflections ... 66 

86. To lay out a parabola by middle ordinates . . . .67 

87. To draw a tangent to a parabola.67 

89. To lay out a parabola by bisecting tangents . . . .68 

90. To lay out a parabola by intersections «... 69 

Article II. — Radius of Curvature. 

92. Definition .... ... .71 

93. To find the radius of curvature at certain stations . . .71 

95. Simplification when the tangents are equal ... 76 




TABLE OF CONTENTS. xiil 

CHAPTER III. 

LEVELLING. 

Article I. — Heights and Slope Stakes. 

SBCT. PAGE 

96. Definitions . . 78 

97. To find the difference of level of two points . . . .78 

98. Datum plane. 79 

99. To find the heights of the stations on a line . . . . 8C 

100. Sights denominated plus and minus .81 

101. Eorm of field notes . . . . . . . .82 

102. To set slope stakes.82 

Article II. — Correction for the Earth’s Curvature and 
for Refraction. 

103. Earth’s curvature.84 

104. Refraction.84 

105. To find the correction for curvature and refraction . . 85 

Article III. —Vertical Curves. 

106. Manner of designating grades . . .86 

107. To find the grades for a vertical curve at whole stations 86 

109. To find the grades for a vertical curve at sub-stations . 88 

Article IV. — Elevation of the Outer Rail on Curves. 

110. To find the proper elevation of the outer rail , 89 

.11. Coning of the wheels.89 

CHAPTER IV. 

EARTH-WORK. 

Article I. — Prismoidal Formula. 

112. Definition of a prismoid.92 

113. To find the solidity of a prismoid.92 

Article II. -Borrow-Pits. 

114. Manner of dividing the ground.93 









XIV 


TABLE OF CONTENTS. 


BECT. _ PASS 

115. To find the solidity of a vertical prism whose horizontal sec¬ 

tion is a triangle.. .93 

116. To find the solidity of a vertical prism whose horizontal sec¬ 

tion is a parallelogram.94 

117. To find the solidity of a number of adjacent prisms having 


the same horizontal section. 

Article III. — Excavation and Embankment. 

A. Centre Heights alone given. 

119. To find the solidity of one section.97 

120. To find the solidity of any number of successive sections . 98 

B. Centre and Side Heights given. 

121. Mode of dividing the ground.99 

122. To find the solidity of one section.100 

123. To find the solidity of any number of successive sections . 104 

125. To find the solidity when the section is partly in excavation 

and partly in embankment .... . . 105 

126. Beginning and end of an excavation ... .107 

C. Ground very Irregular. 

127. To find the solidity when the ground is very irregular . 108 

128. Usual modes of calculating excavation. 109 

D. Correction in Excavation on Curves. 

129. Nature of the correction. 110 

130. To find the correction in excavation on curves . . . 112 

132. To find the correction when the section is partly in excava 

tion and partly in embankment. 113 


TABLES. 

* 

NO* PAG® 

I. Radii, Ordinates, Tangent and Chord Deflections, and Or¬ 

dinates for Curving Rails .115 

II. Long Chords.-» . .119 





TABLE OF CONTENTS, 


X? 

NO. PAGE 

III. Correction for the Earth’s Curvature and for Refraction . 120 
IY. Elevation of the Outer Rail on Curves . . . .120 

Y. Prog Angles, Chords, and Ordinates for Turnouts . .121 

VI, Length of Circular Arcs in Parts of Radius . . . 121 

VJI. Expansion by Heat.122 

VIII. Properties of Materials.123 

IX. Magnetic Variation.126 

X. Trigonometrical and Miscellaneous Eormulce . . 133 

XI, Squares, Cubes, Square Roots, Cube Roots, and Recip¬ 
rocals . . . 137 

XII. Logarithms of Numbers . .... 155 

XIII. Logarithmic Sines, Cosine?; Tangents, and Cotangents 171 

XIV. Natural Sines and Cosines.219 

XV. Natural Tangents and Cotangents . . . 229 

XVL Rise per Mile of Various Grades .... 242 




EXPLANATION OF SIGNS 


The sign -|- indicates that the quantities between which it is placed 
sire to be added together. 

The sign — indicates that the quantity before which it is placed 
Is to be subtracted. 

The sign X indicates that the quantities between which it is placed 
are to be multiplied together. 

The sign -f- or : indicates that the first of two quantities between 
which it is placed is to be divided by the second. 

The sign = indicates that the quantities between which, it is placed 
are equal. 

The sign oo indicates that the difference of the two quantities be* 
tween which it is placed is to be taken. 

The sign stands for the word “hence ” or “ therefore.” 

The ratio of one quantity to another may be regarded as the quo* 
tient of the first divided by the second. Hence, the ratio of a to 6 is 
expressed by a : b, and the ratio of c to d by c : d. A proportion ex 
presses the equality of two ratios. Hence, f. proportion is represented 
by placing the sign = between two ratios; as , a : b = c : d. 


In the text and in the tables the foot has been taken as the unit oi 
measure when no other unit is specified. 






FIEL 


CHAPTER 

CIRCULAR CURVES. 


Article I.— Simple Curves, 


1. The railroad curves here considered are either Circular or Para, 
bolic. Circular curves are divided into Simple, Reversed, and Com¬ 
pound Curves. We begin with Simple Curves. 

2. Let the arc ADEFB (fig. 1) represent a railroad curve, unit- 







2 


CIRCULAR CURVES. 


ing the straight lines G A and B H. The length of such a curve is 
measured by chords, each 100 feet long.* Thus, if the chords AD , 
DE, E F, and FB are each 100 feet in length, the whole curve is 
said to be 400 feet long. The straight lines GA and B H are always 
tangent to the curve at its extremities, which are called tangent points. 
If G A and B H are produced, until they meet in C, A C and B 0 
are called the tangents of the curve. If A C is produced a little beyond 
C to K , the angle KCB, formed by one tangent with the other pro¬ 
duced, is called the angle of intersection , and shows the change of direc¬ 
tion in passing from one tangent to the other. 

The following propositions relating to the circle are derived from 
Geometry. 

I. A tangent to a circle is perpendicular to the radius drawn through 
the tangent point. Thus, A C is perpendicular to A O , and B C to 
BO. 

II. Two tangents drawn to a circle from any point are equal, and if 
a chord be drawn between the two tangent points, the angles between 
this chord and the tangents are equal. Thus A C = B C, and the 
angle B AC = A B C. 

III. An acute angle between a tangent and a chord is equal to half 
the central angle subtended by the same chord. Thus, CAB — 
hAOB. 

TV. An acute angle subtended by a chord, and having its vertex in 
the circumference of a circle, is equal to half the central angle sub¬ 
tended by the same chord. Thus, DAE = \ DOE . 

Y. Equal chords subtend equal angles at the centre of a circle, and 
also at the circumference, if the angles are inscribed in similar seg¬ 
ments. Thus, A 0 D = D O E, and D A E — E A F. 

YI. The angle of intersection of two tangents is equal to the cen¬ 
tral angle subtended by the chord which unites the tangent points. 
Thus, KCB = AOB. 

3. In order to unite two straight lines, as GA and BH, by a curve, 
the angle of intersection is measured, and then a radius for the curve 
may be assumed, and the tangents calculated, or the tangents may be 
assumed of a certain length, and the radius calculated. 


* Some engineers prefer a chain 50 feet in length, and measure the length cf a 
curve by chords of 60 instead of 100 feet. The chord of 100 feet has been adopted 
throughout this article; hut the formulae deduced may he very readily modified to 
Buit chords of any length. See also § 13. 




.981 4 


SIMPLE CURVES. 


3 


4. Problem. Given the angle of intersection KCB = 1 (Jig 1) 
and the radius A 0 = R, to find the tangent A C — T . 



= tan. h 1) 

. •. T = R tan. £ l. 


Example. Given 1 = 22° 52', and R = 3000, to find T. Here 

R = 3000 3.477121 

£ / == 11° 26' tan. 9.305869 

T— 606.72 2.782990 

5. Problem. Given the angle of intersection KCB = I (fig. 1 ), 
md the tangent A C = T, to find the radius AO =*R. 








4 


CIRCULAR CURVES. 


Solution. In the right triangle A 0 G we have (Tab. X. 6) 
12 = cot. AOC, or 1 = cot. h I ; 

Ei 53 R = Tcot. i I. 

Example. Given 7 = 31° J6' and T= 950, to find 72. Here 

T= 950 2.977724 

£ 7= 15° 38 cot. 0.553102 

72 = 3394.89 3.530826 

6. The decree of a curve is determined by the angle subtended at 
its centre by a chord of 100 feet. Thus, if A 0 D = 6° (fig. 1), 
ADEFB is a 6° curve. 

7. The deflection angle of a curve is the acute angle formed at any 
point between a tangent and a chord of 100 feet. The deflection angle 
is, therefore (§ 2, III ), half the degree of the curve. Thus, CAD or 
CBF is the deflection angle of the curve ADEFB , and is half 
AOD or half FOB. 


A. Method by Deflection Angles. 

8. The usual method of laying out a curve on the ground is by 
means of deflection angles. 

9. Problem. Given the radius A 0 = 72 {fig. 1), to find the de¬ 
flection angle C B F — D. 

Sohition. Draw O L perpendicular to BF. Then the angle BOB 
= ^B 0 F = D, and B L = \ B F = 50. But in the right triangle 

OBL we have (Tab. X. 1) sin. B OL = 

^ sin. £> = — 

Example. Given 72 = 5729.65, to find D. Here 

50 1.698970 

72 = 5729.65 3.758128 

D = 30' sin. 7.940842 

Hence a curve of this radius is a 1° curve, and its deflection angle is 
30'. 

10. Problem. Given the deflection angle CBF—D {fig. 1), in 
find the radius A 0 — 72. 




METHOD 33Y DEFLECTION ANGLES. 


50 

Solution. By the preceding section we have sin. D= —, whence 

R 


R sin. D ~ 50 ; 


'. R=-r 


50 


sin. D 


By this formula the radii in Table I. are calculated. 

Erampfo. Given D = 1°, to find R. Here 

50 

D= 1 ° 

R = 2864.93 


1.698970 
sin. 8 241855 


3.457115 


11. Problem. Given the angle of intersection KCB = 7 (fig. 1), 
and the tangent A C = T, to find the deflection angle C A D = D. 

50 

Solution. From § 9 we have sin. D = —, and from § 5, R = 

T cot. £ I. Substituting this value of R in the first equation, we get 

. ~ 50 

Bin. D = 


T cot. i / ’ 


.*. sin. Z> = 


50 tan. % 1 
T 


Example. Given 1 — 21° and T = 424.8, to find D. Here 


50 

1.698970 

\1 = 10° 30 

tan. 9.267967 


0.9669S7 

2’= 424 8 

2.628185 

D = 1° 15' 

sin. 8.338752 


12. Problem. Given the angle of intersection K CB = I (fig. 1) ; 

and the deflection angle CA D ='D, to find the tangent AC — T. 

, . _ 50 tan. £ 1 

Solution. From the preceding section we have sm. D — ———* 

Hence, Tsin. D — 50 tan. \ I‘, 




.-. T = 


50 tan. i 7 
sin. D 


Example. Given 1 = 28° and D = 1°, to find T. Here 


r = 714.31. 


sin l 3 












6 


CIRCULAR CURVES. 


13. Frol>Iem. Given the angle of intersection K CB = 1 (Jig. 1), 
and the dejlection angle C A D = D, to Jind the length of the curve. 

Solution. By § 2 the length of a curve is measured by chords of 100 
feet applied around the curve. Now the first chord A D makes with 
the tangent A C an angle CA D = D, and each succeeding chord 
DE, EF, &c. subtends at A an additional angle DAE, EAF, &c. 
each equal to D; since each of these anglps (§ 2, IV.) is half of a 
central angle subtended by a chord of 100 feet. The angle CAB — 
h A 0 B — % I is, therefore, made up of as many times D, as there arc 
chords around the curve. Then if n represents the number of chords, 
we have nD —\I\ 



If D is not contained an even number of times in ^ I, the quotient 
above will still give the length of the curve. Thus, in fig. 2, suppose 
D is contained 4| times in h I- This shows that there will be four 
whole chords and § of a chord around the curve from A to B. The 
angle GAB, the fraction of D, is called a sub-deflection angle, and 
G B , the fraction of a chord, is called a sub-chord .* 

The length of the curve thus found is not the actual length of the 
arc, but the length required in locating a curve. If the actual length 
of the arc is required, it may be found by means of Table YI. 

Example. Given / = 16° 52' and D — 1° 20', to find the length of 

, tt hi 80 26' 506 ' „ _ _ . , 

the curve. Here n—.~ = .-r—- — — ■ = 6.325, that is, the curve 

JD olr 

is 632.5 feet long. 

To find the arc itself in this example, we take from Table VI. the 
length of an arc. of 16° 52', since the central angle of the whole curve 
is equal to I (§ 2, YI.), and multiply this length by the radius of the 
curve. 

Arc 10° = .1745329 

“ 6° = .1047198 

“ 50' = .0145444 

“ 2' = .0005818 

“ 16° 52' = .2943789 


* This method of finding the length of a sub-chord is not mathematically accu¬ 
rate ; for, by geometry, angles inscribed in a circle are proportional to the arcs on 
which they stand ; whereas this method supposes them to be proportional to the 
chords of these arcs. In railroad curves, the error arising from this supposition w 
too small to be regarded. 




METHCD BY DEFLECTION ANGLES. 


1 


The radius of the curve is found from Table I. to be 2148.79, and this 
multiplied by .2943789 gives 632.558 feet for the length of the arc. 


14. P&'oMem. Given the deflection angle D, to lay out a curve 
from a given tangent point. 



Solution. Let A (fig. 2) be the given tangent point in the tangent 
H C. Set the instrument at A , and lay off the given deflection angle 
D from A C. This will give the direction AD, and 100 feet being 
measured from A in this direction, the point D will be determined 
Lay off in succession the additional angles DAE, E A F, &c., each 
equal to Z>, and make D E, E F, &c. each 100 feet, and the points 
E, F, &c. will be determined. The points D , E , F, &c., thus deter¬ 
mined, are points on the required curve (§ 7, and § 2, III., IV.), and 
are called stations. 

If there is a sub-chord at the end, as G B , the sub-deflection angle 
GAB must be the same part of D that G B is of a whole chord (§ 13). 

15. It is often impossible to lay out tho whole of a curve, without 
removing the instrument from its first position, either on account of 
the great length of the curve, or because some obstruction to the sight 
may be met with. In this case, after determining as many stations as 
possible, and removing the instrument to the last of these stations, we 
ought to be able to find the tangent to the curve at this station; for 








8 


CIRCULAR CURVES. 


then the curve could be continued by deflections from the new tangent, 
in precisely the same way as it was begun from the first tangent. 

16 . Problem. After running a curve a certain number of stations, 
to find a tangent to the curve at the last station. 

Solution. Suppose that the curve (fig. 2) has been run three stations 
to F, and that FL is the tangent required. Produce A F to K, and 
we have the angle KFL = A F C. But (§ 2, II.) AF C = FA C. 
Therefore KFL= FA C. Now FA C is the sum of all the deflec¬ 
tion angles laid off from the tangent at A, that is, in this case, FA C 
— 3D, and the tangent FL is, therefore, obtained by laying off from 
A F produced an angle KFL equal to the total deflection from the 
preceding tangent. 

If the curve is afterwards continued beyond F , as, for instance, to B 
a tangent B N at B is obtained by laying off from FB produced an 
angle MB N = L B F = L F B, the total deflection from the pre 
ceding tangent FL. 

B. Method by Tangent and Chord Deflections. 


17. Let AB CD (fig. 3) be a curve between the two tangents E A 
and D L, having the chords A B, B C, and CD of the same length 



El 

Produce the tangent E A, and from B draw B G perpendicular to 
A G. Produce also the chords A B and B C, and make the produced ' 





METHOD BY TALENT AND CHORD DEFLECTIONS. 9 


parts BH and CK of the same length as the chords. Draw C11 
and D K. B G is called the tangent deflection, and CH or DK the 
ch^rd deflection. 


18. Problem. Given the radius AO = R {fig. 3), to find the 
tangent deflection B G, and the chord deflection C H. 

Solution. The triangle C BII is similar to BOG ; for the angle 
BO C — 180°— {OBC + B CO), or, since B C 0 = A B 0, B 0 C 
= 180° — (0 B C -f- ABO) — CB H, and, as both the triangles are 
isosceles, the remaining angles are equal. The homologous sides are, 
therefore, proportional, that is, B 0 : B C = B C : C H, or, represent¬ 
ing the chord by c and the qhord deflection by d, R : c = c : d ; 



To find the tangent deflection, draw BM to the middle of C11 y 
bisecting the angle CBH , and making BMC a. right angle. Then 
the right triangles BMC and AG B are equal; for B C = AB, and 
the angle C B M = h CB H = i B 0 C = £ A O B = B A G (§2, 
III.). Therefore B G = CM = A CH = £ c?, that is, the tangent de¬ 
flection is half the chord deflection. 


19. Problem. Given the deflection angle D of a curve , to find the 
chord deflection d. 

Solution. By the preceding section we have d = and by $ 10, 
R = gjj—jrj- Substituting this value of R in the first equation, we find 


d = 


c 2 sin. D 


50 


This formula gives the chord deflection for a chord c of any length 
though D is the deflection angle for a chord of 100 feet (§ 7). When 
c = 100, the formula becomes d = 200 sin. Z), or for the tangent de¬ 
flection ^d = 100 sin. D. By these formula the tangent and chord 
deflections in Table I. may be easily obtained from the table of natural 
sines. 


20. The length of the curve may be found by first finding D (§ 9 or 
§11), and then proceeding as in § 13. 


21. Problem. To draw a tangent to the curve at any station , 
tis B {fig. 3). 

Solution. Bisect the chord deflection H C of the next station in M. 
2 



10 


CIRCULAR CURVES. 


A line drawn through B and M will be the tangent required; for it 
has been proved (§ 18) that the angle C B M is in this case equal to 
\ B 0 C, and B M is consequently (§ 2, III.) a tangent at B. 

If B is at the end of the curve, the tangent at B may be found with¬ 
out first laying off H C. Thus, if a chain equal to the chord is extend¬ 
ed to H on A B produced, the point H marked, and the chain then 
swung round, keeping the end at B fixed, until HM = k d, BM will 
be the direction of the required tangent.* 


22. Problem. Given the chord deflection d, to lay out a curve 
from a given tangent point. 

Solution. Let A (fig. 3) be the given tangent point, and suppose d 
has been calculated for a chord of 100 feet. Stretch a chain of 100 
feet from A to G on the tangent EA produced, and mark the point 
G. Swing the chain round towards A B, keeping the end at A fixed, 
until B G is equal to the tangent deflection k d , and B will be the first 
station on the curve. Stretch the chain from B to H on A B pro 
duced, and having marked this point, swing the chain round, until IIG 
is equal to the chord deflection d. C is the second station on the curve. 
Continue to lay off the chord deflection from the preceding chord pro¬ 
duced, until the curve is finished. 

Should a sub-chord DF occur at the end of the curve, find the tan 
gent DL at D (§ 21), lay off from it the proper tangent deflection LI 
for the given sub-chord, making DF of the given length, and F will 
be a point on the curve. The proper tangent deflection for the sub¬ 
chord may be found thus. Represent the sub-chord by c', and the cor¬ 
e's 

responding chord deflection by d\ and we have (§ 18) \ d' = — ; but 


since id = —^ we have kd': ^d = c ' 2 : c 2 . Therefore ^d' = frd 


C / \ 2 


Example. Given the intersection angle 1 between two tangents 
equal to 16° 30', and R = 1250, to find T, d , and the length of the 
curve in stations. Here 

(§4) T=R tan. h I = 1250 tan. 8° 15' = 181.24 ; 

100 2 


(§18) d = — = 

; R 1250 


8 , 


* The distance B M is not exactly equal to the chord, but the error arising from 
taking it equal is too small to be regarded in any curves but those of very small 
radius. If necessary, the true length of B M may be calculated; for B M = 
/Am* — H aB 





ORDINATES. 


n 


(§ 9) sin. D = ^ ^ = .04 =■ nat. sin. 2° 17£'; 


(§ 13) n 


ij __ 8Q 15' 
D 2o 17*' 


495' 

137.5' 


3.60. 


These results show, that the tangent point A (fig. 3) on the first tan 
gent is 181.24 feet from the point of intersection, — that the tangent 
deflection GB = = 4 feet, — that the chord deflection HC or KD 

— 8 feet,— and that the curve is 360 feet long. The three whole sta¬ 
tions B, C, and D having been found, and the tangent DL drawn, the 
tangent deflection for the sub-chord of 60 feet will be, as shown above, 

kd' = 4 (^ q ) 2 = 4 X -6 2 = 4 X .36 = 1.44. LF= 1.44 feet being 


laid off from DL, the point F will, if the work is correct, fall upon 
the second tangent point. A tangent at F may be found (§21) by 
producing DF to P , making FP = D F = 60 feet, and laying off 
PN = 1.44 feet. FN will be the direction of the required tangent, 
which should, of course, coincide with the given tangent. 

23. Curves may be laid out with accuracy by tangent and chord 
deflections, if an instrument is used in producing the lines. But if an 
instrument is not at hand, and accuracy is not important, the lines may 
be produced by the eye alone. The radius of a curve to unite two 
given straight lines may also be found without an instrument by § 73, 
or, having assumed a radius, the tangent points may be found by § 74. 


C. Ordinates. 

24. The preceding methods of laying out curves determine points 
TOO feet distant from each other. These points are usually sufficient 
for grading a road ; but when the track is laid, it is desirable to have 
intermediate points on the curve accurately determined. For this pur¬ 
pose the chord of 100 feet is divided into a certain number of equal 
parts, and the perpendicular distances from the points of division to 
the curve are calculated. These distances are called ordinates. If the 
chord is divided into eight equal parts, we shall have points oh the 
curve at every 12.5 feet, and -this will be often enough, if the rails, 
which are seldom shorter than 15 feet, have been properly curved 
(§ 28). 


25. Problem. Given the deflection angle D or the radius R of a 
curve , to find the ordinates for any chord. 

Solution. I. To find the middle ordinate. Let ABB (fig. 4) be 
a portion of a curve, subtended by a chord A B , which may be de- 


CIRCULAR CURVES. 


i2 

noted by c. Draw the middle ordinate ED, and denote it by m. Pro¬ 
duce ED to the centre F, and join A F and A E. Then (Tab. X. 3' 


E 



— = tan. E AD, or ED = AD tan. E AD. But, since the angle 

E AD is measured by half the arc BE, or by half the equal arc A E , 
we have E AD — k AFE. Therefore E D — AD tan. £ A FE, or 
H3F* m — \ c tan. ^ A FE. 

When c = 100, AFE = D (§ 7), and m — 50 tan. h, D, whence in 
may be obtained from the table of natural tangents, by dividing tan. 
^ D by 2, and removing the decimal point two places to the right. 

The value of m may be obtained in another form thus. In the 
triangle ADFwe have DF= A F 2 — AD* ^ */R* — 4c 2 - Then 
7 n = E F — D F = R — D F, or 

^ m = R — \ZR* — ic 2 . 

II. To find any other ordinate, as RN, at a distance DN = b from 
the centre of the chord. Produce RN until it meets the diameter 
parallel to A B in G, and join R F. Then RG — */RF* — FG* = 

\/R 2 — b*, and RN— RG — NG — R G — DF. Substituting the 
value of R G and that of DF found above, we have 


RN=*/R* — b* — */R* 

















ORDINATES. 


13 


By these formulas the ordinates in Table I. are calculated. 

The other ordinates may also be found from the middle ordinate by 
the following shorter, but not strictly exact method. It is founded on 
the supposition, that, if the half-chord BD be divided into any number 
of equal parts, the ordinates at these points will divide the arc E B into 
the same number of equal parts, and upon the further supposition, that 
the tangents of small angles are proportional to the angles themselves. 
ThSse suppositions give rise to no material error in finding the ordi¬ 
nates of railroad curves for chords not exceeding 100 feet. Making, 
for example, four divisions of the chord on each side of the centre, and 
joining A R, AS, and A T, we have the angle RAN— \EAD, 
since RB is considered equal to | E B. But E A D = i A“FE. 
Therefore, R AN— f A FE. In the same way we should find S A 0 
= ^ A FE, and TAP \ A FE. We have then for the ordinates, 
RN = AN tan. RAN—^c tan. § A FE, S 0 = A 0 tan. SA 0 — 
$ c tan. i A FE, and TP — AP tan. TA P — l c tan. £ A FE. 
But, by the second supposition, tan. \AFE — | tan. £ AFE, 
tan. i A FE — | tan. \ A FE, and tan. J A FE — £ tan. % A FE. 
Substituting these values, and recollecting that ^ c tan. \ AFE — m , 
we have 


RN=^X § c tan. frA FE =* - m, 
S 0 = j X j c tan. £ AFE = —m, 
TP = ^ X f c tan. ^ A FE = ^ m. 


16 


In general, if the number of divisions of the chord on each side of 
the centre is represented by n, we should find for the respective ordi- 

, . . . , (» + 1) (» — 1) m (n-f2)(»-2)m 

nates, beginning nearest the centre, ——~ 2 -» 

(«4-3)(» — 3)m 
'- z* -, &c. 


Example . Find the ordinates of an 8 ° curve to a chord of 100 feet. 

15 3 

Here m = 50 tan. 2 ° = 1.746, RN = j$ m = 1.637, S 0 = 4 m = 1.310, 
and TP = ^ m = 0.764. 

26. An approximate value of m also may be obtained from the for¬ 
mula m — R — JR? — ± c 2 . This is done by adding to the quantity 

£ 4 

under the radical the very small fraction 34 > making it a perfect 







14 


CIRCULAR CURVES. 


iquare, the root of which will be R — 



IS * 3 


. •. m = 



We have, then, m 


R 


27. From this value of m we see that the middle ordinates of any 
two chords in the same curve are to each other nearly as the squares 
of the chords. If, then, A E (fig. 4) be considered equal to h A B, its 
middle ordinate CII — \ED. Intermediate points on a curve may, 
therefore, be very readily obtained, and generally with sufficient accu¬ 
racy, in the following manner. Stretch a cord from A to B , and by 
means of the middle ordinate determine the point E. Then stretch 
the cord from A to E, and lay off the middle ordinate CII — \ ED, 
thus determining the point ( 7 , and so continue to lay off from the suc¬ 
cessive half-chords one fourth the preceding ordinate, until a sufficient 
number of points is obtained. 


P. Curving Rails. 

28. The rails of a curve are usually curved before they are laid. To 
do this properly, it is necessary to know the middle ordinate of the 
curve for a chord of the length of a rail. 


29. Pt'Ohlcin. Given the radius or deflection angle of a curve , to 
find the middle ordinate for curving a rail of given length. 

Solution. Denote the length of the rail by l, and we have (§ 25) 
the exact formula m = R — R 2 — 4 ^ and (§ 26) the approximate 
formula 




m 


kl 

2 R' 


This formula is always near enough for chords of the length of a rail • 

60 

If we substitute for R its value (§ 10) R = gin > we have, 


m = X 


sin. D 
100 


Example. In a 1° curve find the ordinate fora rail of 18 feet m 
length. Here R is found by Table I. to bo 5729.65, and therefore, 





REVERSED AND COMPOUND CURVES. 


15 


by the first formula, m = = .00707. By the second formula, 

m = .81 sin. 30 f = .00707. The exact formula would give the same 
result even to the fifth decimal. 

By keeping in mind, that the ordinate for a rail of 18 feet in a 1° 
curve is .007, the corresponding ordinate in a curve of any other de¬ 
gree may be found with sufficient accuracy, by multiplying this deci¬ 
mal by the number expressing the degree of the curve. Thus, for a 
curve of 5° 36' or 5.6°, the ordinate would be .007 X 5.6 = .039 ft. = 
.468 in. 

For a rail of 20 feet we have \l 2 — 100, and, consequently, m — 
sin. D. This gives for a 1° curve, in — .0087. The corresponding or¬ 
dinate in a curve of any other degree may be found with sufficient 
accuracy, by multiplying this decimal by the number expressing the 
degree of the curve. 

By the above formula for m ) the ordinates for curving rails in Table 
I, are calculated. 


Article II. — Reversed and Compound Curves. 

30. Two curves often succeed each other having a common tangent 
at the point of junction. If the curves lie on opposite sides of the com¬ 
mon tangent, they form a reversed curve, and their radii may be the 
same or different. If they lie on the same side of the common tangent, 



they have different radii, and form a compound curve. Thus ABC 
(fig. 5) is a reversed curve, and AB D a compound curve. 



16 


CIRCULAR CURVES. 


31. Problem. To lay out a reversed or a compound curve, tvheit 
the radii or deflection angles and the tangent points are known. 

Solution. Lay out the first portion of the curve from A to B ffig. 5), 
by one of the usual methods. Find B F, the tangent to A B, at the 
point B (§ 16 or § 21). Then B F will be the tangent also of the sec¬ 
ond portion B C of a reversed, or B D of a compound curve, and from 
this tangent either of these portions may be laid off in the usual man 
ner 


A. Reversed Curves. 

32 Theorem. The reversing point of a reversed curve between 
parallel tangents is in the line joining the tangent points. 



Demonstration. Let A CB (fig. 6 ) be a reversed curve, uniting the 
parallel tangents HA and B K , having its radii equal or unequal, and 
reversing at C. If now the jhords A C and CB are drawn, we have 
to prove that these chords are in the same straight line. The radii 
E C and C F, being perpendicular to the common tangent at C (§ 2 ,1.), 
are in the same straight line, and the radii A E and B F, being per¬ 
pendicular to the parallel tangents H A and B K, are parallel. There¬ 
fore, the angle A E C = CFB, and, consequently, E C A, the half 
supplement of A E C,\ is equal to F C B, the half supplement of CFB ; 
but these angles cannot be equal, unless A C and CB are in the same 
straight line. 

33. Problem. Given the perpendicular distance between two par¬ 
allel tangents B D = b {fig. 6 ), and the distance between the two tangent 
points A B = a, to determine the reversing point C and the common radius 
EC— C F — R of a reversed curve uniting the tangents HA and B K. 

Solution. Let ACB be the required curve. Since the radii are 






REVERSED CURVES. 11 

equal, and the angle A E C = B F C, the triangles AE C and BFC 
are equal, and AC — C B = £ a. The reversing point C is, therefore , 
the middle point of A B. 

To find R, draw E G perpendicular to A C. Then the right tri¬ 
angles AEG and BAD are similar, since (§ 2, III.) the angle 
BAD — % A EC = AEG. Therefore A £ : A G — A B : B D, 
or R: \ a — a :b\ 

13^ .-.R== “1. 

4 b 

Corollary. If R and b are given, to find a, the equation R — ~ 
gives a 2 = 4 Rb; 

ESP* . •. a = 2 J~RT. 

Examples. Given b = 12 , and a — 200, to determine R. Here 
R _ _ ioooo _ 

n ~ 4 X 12 ~ 12 “ 

Given R — 675, and b = 12, to find a. Here a — 2^/675 X 12 = 
2^8100 == 2 X 90 = 180. 

34. Problem. Given the perpendicular distance between two par¬ 
allel tangents B D — b {fig. 7), the distance between the two tangent points 
A B = a, and the first radius E C — R of a reversed curve uniting the 
tangents HA and BK. to find the chords AC — a' and C B = a n , and 
the second radius CF — R'. 



Solution. Draw the perpendiculars E G and FL. Then the right 
triangles ABD and EA G are similar, since the angle BAD =* 









CIRCULAR CURVES. 


18 


\AEC= AEG. Therefore AB : B D = E A : A G, or a : b 

2 Rb 

. ’. a' = —— * 


Since a' and a" are (§ 32) parts of a, Ave haA r e 
a' 1 = a — a 1 . 


To find R' the similar triangles ABB and FB L give AB: BB 
= FB : BL, or a : b = R<: $ a"; 




Example. Given b = 8, a = 160, and R = 900, to find a\ a n , and 

2'. He 
160 X 70 


R'. Here a' = - * ^ X8 = 90, a" = 160 — 90 = 70, and R' = 


2X8 


= 700. 


35. Corollary 1. If 6, a', and a" are given, to find a, 72, and 72-, 
we have (§ 34) 

a = a'+a"; 72 = ^'; 72'= 

2 b 2 b 

Example. Given b — 8, a' = 90, and a' f — 70, to find a, R , and 72 

Here a = 90 -f 70 = 160, R = -f£g- = 900, and R' = = 

700. 


36. Corollary 2. If R , 72', and 6 are gi\ r en, to find a, a\ and a", 
re have (§ 35), R - 
b* = 2 6(72^-72'); 


. -n. i aa , + ««" a (a'-fa'*) a2 

we have (§ 35), R -f~ R 1 — — 26 — = “"26 — = 26* Therefore 


.-.a = ^2 6 (72 + 72'). 
Having found a, avc have (§ 34) 


2726 . 


2 72' 6 


Example. Given R = 900, 72' = 700, and 6 = 8, to find a, a and 
a". Here a = J2 X 8(900 + 700) = ^/16 X 1000 — 160, a' =- 










REVERSED CURVES. 


19 


37. Frol)loill. Given the angle A KB — K, which shows the 
change of direction of two tangents HA and B K {fig. 8), to mite these 
tangents by a reversed curve of given common radius Z2, starting from a giv - 
tn tangent point A. 



Solution. With the given radius run the curve to the point D, where the 
tangent D Nbecomes parallel to B K. The point D is found thus. Since 
the angle NG iT, which is double the angle HAD (§ 2, II.), is to be 
made equal to A KB — K, lay off from HA the angle II AD — i K 
Measure in the direction thus found the chord AD = 2 R sin. % K. 
This will be shown (§ 69) to be the length of the chord for a deflection 
angle \ K. Having found the point D, measure the perpendicular dis¬ 
tance D M = b between the parallel tangents. 

The distance DB=2DC=a may then be obtained from the for¬ 
mula (§ 33, Cor.) 

o = 2 jRb . 

The second tangent point B and the reversing point C are now de¬ 
termined. The direction of D B or the angle B DN may also be ob¬ 
tained ; for sin, BDN = sin. DBM = or 

^ sin. BDN — - . 

a 

38. Problem. Given the line A B — « {fig. 9), which joins the 
fixed tangent points A and B , the angles H A B = A and A B L = B % 
and the first radius A E = R, to find the second radius B F = R 1 of a 
reversed curve to unite the tangents H' A and B K. 

First Solution. With the given radius run the curve to the point D t 
rhere the tangent D N becomes parallel to B K. The point D is found 






20 


CIRCULAR CURVES. 


thus. Since the angle H G N, which is double HAD (§ 2 , II.), is 
equal to A oo B, lay off from HA the angle HAD — % (A<*>B), and 
measure in this direction the chord AD = 2 R sin. £ (A ooB) (§ 69) 



Setting the instrument at D, run the curve to the reversing point C in the 
line from D to B (§ 32), and measure D C and CB. Then the similar 
triangles DEC and B F C give D C: DE = CB : BF, or D C : R 
«= CB:R 

.-.r'=£Exr 

U o 


Second Solution. By this method the second radius may be found 
by calculation alone. The figure being drawn as above, we have, in 
the triangle A B Z), A B = a, AD = 2R sin. % (A — B ), and the 
included angle DAB — HA B — HAD = A — ^ (A — B) = 
^ [A -f- B). Find in this triangle (Tab. X. 14 and 12) B D and the 
angle ABD. Find also the angle DBL — B AB D. 

Then the chord CB = 2 R' sin. % B F C — 2 RJ sin. DBL , and 
the chord D C = 2 R sin. % D E C = 2 R sin. DBL (§ 69). But 
CB — B D — DC; whence 2 R' sin. D B L = B D — 2 R sin 


DBL-, 


. RJ = 


BD 


2 sin. DBL 


R. 


When the point D falls on the other side of A, that is, when the 
angle B is greater than A , the solution is the same, except that the 
angle DAB is then 180°— £ {A -f- B), and the angle DBL — B — 
ABD. 







REVERSED CURVES. 


21 


39. Problem. Given the length of the common tangent D G — a, 
and the angles of intersection I and V {fig. 10), to determine the common 
radius C E = C F — R of a reversed curve to unite the tangents IIA 
and B L. 



Solution. By § 4 we have D C — R tan. | /, and CG = R tan. £ I 
whence R (tan. \ I- f* tan. £ /') = D C -j- C G — a, or 


R = 


a 

tan. | tan. ^ V 


This formula may be adapted to calculation by logarithms; for we 

have (Tab. X. 35) tan. £Z-f- tan - \ — ^o a'^/cos $j » Substituting 

this value, we get 


r, a cos. A 1 cos. A- T 
11 “ sin./ ( / + / )- 


The tangent points A and B are obtained by measuring from D a 
distance A D = R tan. ^ 7, and from G a distance B G = R tan. \ T. 


Example. Given a = 600, 1 — 12°, and V — 8°, to find R. Here 

a = 600 2.778151 

1 1 = 6° cos. 9.997614 

fr = 4° cos. 9.998941 

2.774706 
sin. 9.239670 


j(I+I>) = 10° 

R == 3427.96 


3.535036 









22 


CIRCULAR CURVES. 


40. Problem. Given the line A B — a (Jig. 10), which joins the 
fixed tangent points A and B, the angle DAB = A> and the angle 
AB G = B, to Jind the common radius EC— CF — Rof a revers'd 
curve to unite the tangents H A and B L. 



Solution. Find Jirst the auxiliary angle A KE — B KF, which may 
he denoted by K. For this purpose the triangle AEK gives AE: E K 
— sin. K : sin. EAR. Therefore EK sin. R= AE sin. EAR — 
R cos. A, since EAR — 90° — A. In like manner, the triangle 
B FR gives FR sin. R — BF sin. FBR — R cos. B. Adding 
these equations, we have (E R -f- FR) sin. R = R (cos. A -f- cos. B ), 
or, since E R -f- F R = 2 R, 2 R sin. R = R (cos. A -f- cos. B ) 
Therefore, sin. R — \ (cos. A -f- cos. B). For calculation by loga¬ 
rithms, this becomes (Tab. X. 28) 

CIS 5 " sin. R — cos. \ (A -f- B) cos. £ (A — B). 

Having found R, we have the angle AER — E = 180° — R — 
EA R = 180° — R — (90° — A) = 90° -f A — R, and the angle 
BFR = F= 180° — R — FB R = 180°— R— (90 ° — B) = 90° 
-f - B — R. Moreover, the triangle AE R gives AE s A R — 
6in. R : sin. E , or R sin. E— AR sin. R, and the triangle BFR gives 
BF:BR = sin. R: sin.F 7 , or R sin. F= BR sin. K. Adding these 
equations, we have R (sin. E -J- sin. F) = (AR-\- B R) sin. R — 
b sin. R. Substituting for sin. E -}- sin. jP its value 2 sin. J (E F\ 





COMPOUND CURVES. 


23 


cos. £ (E — F) (Tab. X. 26), we have 2 11 sin. \ (E -f F) cos. 
%{E — F) = a sin. K. Therefore R — sill< ^ ^ cos . %(E—F) * Fi * 

nally, substituting for E its value 90° -f- A — K , and for F its value 
90° -f B — K, we get £ (E F) = 90° — [K — £ ( A -f- 2?)], and 
\ (E — F) = \ ( A — B) ; whence 

22 =_ 2 « s in. K _ 

cos. [K — % (A 4- B )J cos. £ (4 — 2?) 


Example. Given a =1500, A = 18°, and B = 6 °, to find R. Here 
£ (A -f B) = 12 ° cos. 9.990404 

£ (H — 2?) = 6 ° cos. 9.997614 


IC = 76° 36' 10 " 
^ a = 750 


sin. 9.988018 
2.875061 


2.863079 

K — %(A-\-B)= 64° 36' 10 cos. 9.632347 
k iA “ B ) = 6 ° cos. 9.997614 


9.629961 

R = 1710.48 3.233118 


B. Compound Curves. 

41. Theorem. If one branch of a compound curve be produced , 
until the tangent at its extremity is parallel to the tangent at the extremity 
of the second branch , the common tangent point of the two arcs is in the 
straight line produced, which passes through the tangent points of these par¬ 
allel tangents. 

Demonstration. Let A CB (fig. 11 ) be a compound curve, uniting 
the tangents HA and2?2T. The radii CE and C F, being perpen¬ 
dicular to the common tangent at C (§ 2 ,1.), are in the same straight 
line. Continue the curve AC to D , where its tangent OD becomes 
parallel to BK , and consequently the radius DE parallel to B F. 
Then if the chords CD and CB be drawn, we have the angle CED 
— CFB ; whence E C D, the half-supplement of C E D, is equal to 
F CB, the half-supplement of CFB. But E CD cannot be equal to 
F CB, unless CD coincides with CB. Therefore the line BD pro- 
inced passes through the common tangent point C. 








24 


CIRCULAR CURVES. 


42. Problem. To find a limit in one direction of each radius of a 
compound curve. 



Solution. Let AI and B1 (fig. 11) be the tangents of the curve. 
Through the intersection point 7, draw IM bisecting the angle A IB. 
Draw A L and B M perpendicular respectively to A I and B 7, meet¬ 
ing 1M in L and M. Then the radius of the branch commencing on 
the shorter tangent AI must be less than A L , and the radius of the 
branch commencing on the longer tangent BI must be greater than 
B M. Tor suppose the shorter radius to be made equal to A L, and 
make IN = A 7, and join L N. Then the equal triangles AIL and 
NIL give AL — LN", so that the curve, if continued, will pass 
through N, where its tangent will coincide with IN. Then (§ 41) the 
common tangent point would be the intersection of the straight line 
through B and N with the first curve; but in this case there can be no 
intersection, and therefore no common tangent point. Suppose next, 
that this radius is greater than A L, and continue the curve, until its 
tangent becomes parallel to BI. In this case the extremity of the 







COMPOUND CURVES. 


25 


carve will fall outside the tangent BI in the line A N produced, and a 
straight line through B and this extremity will again fail to intersect 
the curve already drawn. As no common tangent point can be found 
when this radius is taken equal to A A or greater than A L , no com¬ 
pound curve is possible. This radius must, therefore, be less than A L. 
In a similar manner it might be shown, that the radius of the other 
branch of the curve must be greater than B M. If we suppose the tan¬ 
gents AI and B 7 and the intersection angle 7 to be known, we have 
(§ 5) A L — A I cot. £ 7, and B M = BI cot. ^ 7. These values are, 
therefore, the limits of the radii in one direction. 

43. If nothing were given but the position of the tangents and the 
tangent points, it is evident that an indefinite number of different com¬ 
pound curves might connect the tangent points; for the shorter radius 
might be taken of any length less than the limit found above, and a 
corresponding value for the greater could be found. Some other con¬ 
dition must, therefore, be introduced, as is done in the following 
problems. 


44 . PB’oMeili. Given the line AB — a (Jig. 11 ), which joins the 
Jixed tangent points A and B , the angle B A 1 — A, the angle AB I = 
B , and tliejirst radius 4£ = i?, to find the second radius B F = R’ of 
a compound curve to unite the tangents HA and B K. 

Solution . Suppose the first curve to be run with the given radius 
from A to 7), where its tangent D 0 becomes parallel to B 7, and 
the angle IA D = \ (A -f- B). Then (§41) the common tangent 
point C is in the line B D produced, and the chord CB = CD -j- 
BD. Now in the triangle ABD we have AB = a, AD — 272 
sin. ^ (A -f- B) (§ 69), and the included angle DAB = I AB — 
IA D = A — £ (A 4 - B) = £ (A — B). Find in this triangle 
(Tab. X. 14 and 12 ) the angle AB D and the side B D. Find also the 
angle CBI= B — ABD. 

Then (§ 69) the chord CB = 2 R> sin. CB 7, and the chord CD = 
2 R sin. CD 0 = 2 R sin. CB I. Substituting these values of CB 
and CD in the equation found above, CB = CD -f- B D, we have 
2R' sin. CBI=2R sin. CBI+BD) 


. R' = R + 


B D 


2 sin. CB I 


When the angle B is greater than A, that is, when the greater radius 
is given, the solution is the same, except that the angle D A B =* 



26 


CIRCULAR CURVES. 


J (Z? — A), and CBI is found by subtracting the supplement of A Li D 

from B. We shall also find CB = CD — BD, and consequently 

Rt — R _—— 

2 sin .CBI' 

If more convenient, the point D may be determined in the field, by 
laying off the angle IA D — % (A -f- B ), and measuring the distance 
A D = 2 R sin. ^ ( A -f* B). BD and (72?/may then be measured, 
instead of being calculated as above. 


Example. Given a = 950, A = 8°, B = 7°, and R = 3000, to find 
R'. Here A D = 2 X 3000 sin. 1 (8° -f 7°) = 783.16, and DA B — 
£ (8° — 7°) = 30'. Then to find A B D we have 


A B — A D — 166.84 

2.222300 

£ (ADB -f ABD) = 89° 45' 

tan. 2.360180 


4.582480 

AB -f AD = 1733.16 

3.238839 

£ {ADB — ABD) = 87° 24' 17" 

tan. 1.343641 

.-.ABD = 2° 20' 43" 


Next, to find B D t 


AD — 783.16 

2.893849 

DAB = 30' 

sin. 7.940842 


0.834691 

ABD = 2° 20' 43" 

sin. 8.611948 

BD = 167.01 

2.222743 

B — ABD = CBI = 4° 39' 17" 

sin. 8.909292 

2 ( R ' — R) = 2058.03 

3.313451 


.'.R' — R = 1029.01 
.'.Rt = 3000 4- 1029.01 = 4029.01 

To find the central angle of each branch, we have Cl B = 2 C B1 
= 9° 18' 34", which is the central angle of the second branch; and 
AEC = AED — CED = A+ B — 2CB1= 5° 41' 26", which 
is the central angle of the first branch. 

45. Problem. Given {jig. 11) the tangents A1 — T, B1 = T' y 
the angle of intersection = I, and the first radius A E = R, to find the 
second radius B F = R'- 

Solution. Suppose the first curve to be run with the given radius 
from A to D, where its tangent D 0 becomes parallel to BI. Through 







COMPOUND CURVES. 


27 


D draw D P parallel to A R and v/o have IP = DO = AO = 
R tan. ^ 7 (§ 4). Then in the triangle DPB we have D P — 10 =» 
^7—^0 = T-r R tanA R BP = B I-IP =Tf—R tan. % I, 
and the included angle D P B = A IB — 180° — 7. TTnc? in f/u's tri- 
angle the angle GBR and the side BD. The remainder of the solution 
is the same as in § 44. The determination of the point D in the field 
is also the same, the angle I AD being here = ^ 7. When B is 
greater than A, that is, when the greater radius is given, the solution is 
the same, except that DP — R tan. \I — T\ and B P = R tan. \ 1 

— T 1 . 

Example. Given T = 447.32, T f = 510.84, 7=15°, and R = 3000, 
to find R'. Here R tan. ^7 = 3000 tan. 7^° = 394.96, DP — 447.32 

— 394.96 = 52.36, B P = 510.84 — 394.96 = 115.88, and D P B = 
180° — 15° = 165°. Then (Tab. X. 14 and 12) 


BP — DP = 63.52 

1.802910 

\ (BDP + PBD) = 7° 30' 

tan. 9.119429 


0.922339 

BP -{-DP — 168.24 

2.225929 

% (BDP — PBD) = 2° 50' 44" 

tan. 8.696410 

.-. PBD = CBI= 4° 39' 16" 


Next, to find B D , 


DP= 52.36 

1.719000 

DPB = 15° 

sin. 9.412996 


1.131996 

PBD = 4° 39' 16" 

sin. 8.909266 

BD — 167.005 

2.222730 


lhe tangents in this example were calculated from the example in 
^ 44. The values of CBI and B D here found differ slightly from 
those obtained before. In general, the triangle DBP is of better 
form for accurate calculation than the triangle ADB. 

46. If no circumstance determines either of the radii, the condition 
may be introduced, that the common tangent shall be parallel to the 
line joining the tangent points. 


Problem. Given the line A B — a (fig. 12), which unites the 
fixed tangent points A and B , the angle 1AB = A, and the angle 
A BI = B, to find the radii A E — R and B F — R' of a 'compound 
turi'e, having the common tangent D G parallel to A B. 






28 


CIRCULAR CURVES. 


Solution. Let A C and B C be the two branches of the required 
curve, and draw the chords A C and B C. These chords bisect the 



angles A and B; for the angle DA C=£ZDG = £ZAB, and the 
angle GB C — \ D GZ = \ ABI. Then in the triangle A CB wo 
have A C: A B = sin. ABC: sin. A C B. But A C B = 180° — 
( CA B -j- CB A) — 180° — ^ (A + ^)> and as the sine of the sup¬ 
plement of an angle is the same as the sine of the angle itself, 
sin. A CB = sin. ^ ( A -f- B). Therefore A C: a = sin. ^ B : sin. 

£ (A -f- B ), or A C = a similar manner we should 

find Now we have (§ 68) R = , and 

R 1 = — ^ , or, substituting the values of AC and B C just found. 

|^r» j> _ h a sin- ^ B . m __ 2 a s in- 2 A 

sin. ^ A sin. ^ ( A -f- B) ’ sin. ^ B sin. £ ( A -f- B) ’ 

Example . Given a — 950, A = 8°, and B = 7°. to find R and R' 
Here 











COMPOUND CURVES. 


29 


£ a = 475 2.676594 

££-=3° 30' sin. 8.785675 


1.462369 

£ A = 4° sin. 8.843585 
i (A 4- B) = 7° 30' sin. 9.115698 

7.959283 

R = 3184.83 3.503086 

according to the formula for R! 

£.676694 
sin. 8.843585 

1.520279 

% B = 3° 30' sin. 8.785675 
£ (A B)= 7° 30' sin. 9.115698 

7.901373 

Ri = 4158.21 3.618906 


Transposing these same logarithms 
«e have 

£ a = 475 
M = 4° 


47. Problem. Given the line A B — a (Jig. 12), which unites the 
jixed tangent points A and B, and the tangents AI — T and BI — T', 
to Jind the tangents AD = x and B G = y of the two branches of a com¬ 
pound curve , having its common tangent D G parallel to A B. 

Solution. Since D C = A D = x, and C G = B G = y, we have 
D G = x -f- y. Then the similar triangles IDG and 1AB give 
ID : IA — D G : A B, or T — x ■ T — x -j- y : a. Therefore 
aT — ax = Tx -f- Ty (1). Also d 9 : A I = B G : B I, or 
x : T — y : T'. Therefore Ty = T r .(**.). Substituting in (1) the 
value of Ty in (2), we have a T — ax T r + x, or a x -f- Tx -f- 
T'x = aT] 

rip* • x — ° ^ 

^ *' a + T+T ' 1 

T'x 

and, since from (2 ),y = —jr , 


BP 


y = 


aT • 

a -J- 2 r -j- T'' 


The intersection points D and G and the common tangent point C 
are now easily obtained on the ground, and the radii may be found by 
the usual methods-. Or, if the angles IA B — A and AB1 — B 










30 


CIRCULAR CURVES. 


have been measured or calculated, we have (§ 5) R = x cot. | A, and 

R f = y cot. f B. Substituting the values of x and y found above, we 

a T cot £ A nr a T< cot. i B 

have It = a j'i > ana B ~ a -j- T -J- T 1 ’ 


Example. Given a = 500, T = 250, and T 1 — 290, to find x and^ 
Here a + T + V = 500 + 250 + 290 = 1040; whence * = 500 X 
250 -f 1040 = 120.19, and y = 500 X 290 -f- 1040 = 139.42. 

43. ProMem. Given the tangents A1 — T, Bl — T\ and tin 
angle of intersection /, to unite the tangent points A and B {fig. 13) by a 
compound curve, on condition that the two branches shall have their angles 
of intersection IDG and I GD equal. 



buiattun. bince ID G = 1 GD = \ 7, we have ID = 1 G. Rep • 
resent the line ID — / G by x. Then if the perpendicular IH be let 


* The radii of an oval of given length and breadth, or of a three-centre arch of given 
span and rise, may also be found from these formulae. In these cases A -j- B = 90°, 

a T 


and the values of R and R 1 may be reduced to R = 
a T< 


and R 1 


a+ T — T >’ 
ealeulated 


a+f-r 

These values admit of an easy construction, or they may be readily 














TURNOTj TS AND CROSSINGS. 


31 


fall from 7, we have (Tab. X. 11) D 77 = 77) cos. ID G — x cos. £ 7, 
and D G = 2 x cos. \1. BxxtDG=;DC+CG = AD + BG = 
T — 2 + T' — x — T - f- T> — 2 x. Therefore 2 x cos. £ 7 = 
T + 7 1 ' — 2x, or 2 ar + 2 r cos. ^ 7 = 7’ -f- T 1 j whence a- = 

l -fip’ or (Tab. X. 25) 

r _ j(T+T>) 
cos. 2 ^ 7 


The tangents AD = T — x and B G = T' — a: are now readily 
found. With these and the known angles of intersection, the radii oi 
deflection angles maybe found (§ 5 or § 11). This method answers 
very well, when the given tangents are nearly equal; but in general 
the preceding method is preferable. 


Example. Given T — 480, T 1 — 500, and I — 18°, to find x. Hera 
a (T-f T') = 245 2.3891G6 

i 7= 4° 30' 2 cos. 9.997318 

a: = 246.52 2.391848 

Then A D = 480 — 246.52 = 233.48, and B G = 500 — 246.52 = 
253.48. The angle of intersection for both branches of the curve being 
9°, we fmd the radii A E — 233.48 cot. 4° 30' = 2P56.65, and B F == 
253.18 cot.. 4° 30' = 3220.77. 


Article III.— Turnouts and Crossings. 

49. The usual mode of turning off from a main track is by switch¬ 
ing a pair of rails in the main track, and putting in a turnout curve 
tangent to the switched rails, with a frog placed where the outer rail 
of the turnout crosses the rail of the main track. A B (fig. 14) repre¬ 
sents one of the rails of the main track switched, B F represents the 
outer rail of the turnout curve, tangent to AB, and F shows the posi¬ 
tion of the frog. The switch angle, denoted by S, is the angle DAB , 
formed by the switched rail A B with A 7), its former position in the 
main track. The frog angle, denoted by F, is the angle GFM made 
by the crossing rails, the direction of the turnout rail at F being the 
tangent FM at that point. In the problems of this article the gauge 
of the track D C. denoted by g, and the-distance DB , denoted by d , 
are supposed to be known. The switch angle S is also supposed to 
be known, since its sine (Tab. X. 1) is equal to d divided by the lengtu 





32 


CIRCULAR CURVES. 


of the switched rail. If, for example, the rail is 18 feet in length and 
d = .42, we have S == 1 ° 20 '. 

A. Turnout from Straight Lines. 

50. Problem. Given the radius R of the centre line of a turnout 
(Jig. 14), to find the frog angle GFM — Fand the chord B F. 



Solution. Through the centre E draw E K parallel to the m in 
track. Draw B H and FK perpendicular to E K , and join E F. 
Then, since E F is perpendicular to FM and FK is perpendicular to 
F G, the angle EFK = GFM — F", and since E B and B H are 
respectively perpendicular to AB and AD , the angle EBH—DAB 

FK 

= S. Now the triangle E FK gives (Tab. X. 2) cos. E FK = -g-jr • 
But E F, the radius of the outer rail, is equal to R -{- £ < 7 , and 
FK = C H = B B — B C = B E cos. EBH — BC — ^R + h 9) 
cos. S — (g — d). Substituting these values, we have cos. E FK = 
{R + j g) cos. S -( g-d) 

R + is ,or 

l^F cos. F — cos. S — ~~ f . 

B + iff 

From this formula F may he found by the table of natural cosines 
To adapt it to calculation by logarithms, we may consider ^ — d to be 
«qual to (g — d) cos. S, which will lead to no material error, since 










TURNOUT FROM STRAIGHT LINES. 


33 


g — d is very small, and cos. S almost equal to unity 
cos. F then becomes 


or cns. F = ( R — 2 9 + d ) cos. S 

R + %y 


The value of 


To find BF , the right triangle B CF gives (Tab. X. 9) B F = 

2 > p 

Bi n. .gir e * But B C — g d and the angle BF C = BFE 
CFE = (90° — %BEF) — (90° — F) = F — % B E F. But 
BEF = B LF — EBL = F— £. Therefore FFtf = F — 
2 - (F— ^) = i (F+ /S'). Substituting these values in the formula 
for B F, we have 


B F = 


<7 — ^ 

sin. (F S) 


By the above formula the columns headed Fand BF in Table V 
are calculated. 


Example. Given g — 4.7, d — .42, S = 1° 20 ', and R = 500, to 
find Fand B F. Here nat. cos. S — .999729, g — d = 4.28, R -f- \g 
= 502.35, and 4.28 502.35 == .008520. Therefore nat. cos. F = 

.999729 — .008520 = .991209, which gives F = 7° 36' 10 ". Next, to 
find BF, 

g — d = 4.28 0.631444 

\ (F + S) — 4° 28' 5" sin. 8.891555 

BF= 54.94 1.739889 


51. Problem. Given the frog angle GFM = F (ySy. 14), /o 
find the radius R of the centre line of a turnout , and the chord B F. 
Solution. From the preceding solution we have cos. F == 

£+Jg) ^S _ Therefore (R + i 9) j. 0S ' F={/t + i9) 

cos. S — (y — c?), or 


^ + 2 9 — 


g — d 


cos. S — cos. F 


For calculation by logarithms this becomes (Tab. X. 29) 


R + 2 9 -— 


%(9-d) 


sin. £-(F+ S) sin. £ (F— S) 

Having thus found R + £ g, we find R by subtracting | g. BF is 
found, as in the preceding problem, by the formula 

B F= _ g ~ d _ 




sin. £ (F + S) 


3 










34 


CIRCULAR CURVES. 


Example. 
E. Here 


Given g — 4.7, d = .42, S — 1° 20', and F = 7°, to find 


H9 — d )= 2.14 
£ (F-f S) = 4° 10' 
£ (,P— £) = 2° 50' 


0.330414 

sin. 8.861283 
sin. 8.693998 


7.555281 


R + \ g = 595.85 
.*. R = 593.5 


2.775133 


52. Problem. To find mechanically the proper position of a given 
frog. 

Solution. Denote the length of the switch rail by l, the length of the 
frog by f, and its width by w. From B as a centre with a radius 
BH = 2l, describe on the ground an arc G HK (fig. 15), and from 
the inside of the rail at G measure G H — 2 d, and from H measure 
IIK such that HK:BH—\w: fi or UK : 2 1 = \w \ f\ that is, 

IIK = y-. Then a straight line through B and the point K will 
strike the inside of the other rail at F , the place for the point of the 



frog. For the angle HB K has been made equal to | F, and if B M 
be drawn parallel to the main track, the angle MBH is seen to be 
equal to S. Therefore, MBK—BFC = % (F -f- S ), and this 
was shown (§ 50) to be the true value of B F C. 

53 . If the turnout is to reverse, and become parallel to the main 
track, the problems on reversed curves already given will in general 
be sufficient. Thus, if the tangent points of the required curve are 
fixed, the common radius may be found by § 40. If the tangent point 
at tfye switch is fixed, and the common radius given, the reversing 
oint and the other tangent point may be found by § 37, the change 
of direction of the two tangents being here equal to S. But when the 









TURNOUT FROM STRAIGHT LINES. 


35 


frog angle is given, or determined from a given first radius, and the 
point of the frog is taken as the reversing point, the radius of the sec¬ 
ond portion may be found by the following method. 

54. Problem. Given the frog angle F and the distance IIB = b 
(fig. 16) between the main track and a turnout , to find the radius R' of the 
second branch of the turnout , the reversing point being taken opposite F t the 
point of the frog. 



Solution. Let the arc FB be the inner rail of the second branch, 
FG = R' — £ g its radius, and B the tangent point where the turnout 
becomes parallel to the main track. Now since the tangent FK is one 
side of the frog produced, the angle HFK = F, and since the angle 
of intersection at K is also equal to F, BFK — ^ F 2, II.); whence 

BFH=>\F. Then « 68) F G = ^§Tk > ° r E' - i g = 

aiii. kF' But BF — si.,, /; /,- jj X. 9), or ) /> F &jn jj,. Sub 
stituting this value of \B F, we have 

»-*>-&&JF 

In measuring the distance HB — b, it is to be observed, that the 
widths of both rails must be included. 














36 


CIRCULAR CURVES. 


Example. 


Given b = G 2 and F = 8 °, to find R'. 

%b = 3.1 
bF= 4° 

±BF= 44.44 
\F=* 4° 

R' — 29 — 637 .Q8 
•. = 639.43 


Here 

0.491362 
sin. 8.843585 

1.647777 
sin. 8.843585 

2.804192 


B. Crossings on Straight Lines. 

55. When a turnout enters a parallel main track by a second switch, 
it becomes a crossing. As the switch angle is the same on both tracks 
a crossing on a straight line is a reversed curve between parallel tan 
gents. Let HD and NK (fig. 17) be the centre lines of two parallej 
tracks, and HA and B K the direction of the switched rails. If now 
the tangent points A and B are fixed, the distance A B = a may be 
measured, and also the perpendicular distance B P — b between the 
tangents HP and B K. Then the common radius of the crossing 
AC B may be found by § 33 ; or if the radius of one part of the cross¬ 
ing is fixed, the second radius may be found by § 34. But if both frog 
angles are given, we have the two radii or the common radius of a 
crossing given, and it will then be necessary to determine the distance 
A B between the two tangent points. 

56. Problem. Given the perpendicular distance GN—b (Jig. 17) 
between the centre lines of two parallel tracks , and the radii E C — R and 
C F — R' of a crossing , to find the chords A C and B C. 

Solution. Draw E G perpendicular to the main track, and A L , 
C M, and B L* parallel to it. Denote the angle A E C by E. Then, 
since the angle A EL = A HG — S, we have CEL = E -{- S, 
and in the right triangle CEM (Tab. X. 2), CE cos. CE M = 
R cos. (E + S) = EM=EL — LM. But EL = AE cos. A EL 
— R cos. S, and L M : L' M = A C : B C. Now A C : B C = 
E C: CF — R : R*. Therefore, L M: D M = R: R', or L M: L M 
+ L'M = R : R + R’i that is, LM: b — 2 d = R : R + R', whence 

L M = ’ Substituting these values of EL and L M in the 

equation for R cos. (E + S), we have R cos. (E + S) — R cos. S — 
R (b — 2d) 

R+R' » 






CROSSINGS ON STRAIGHT LINES. 


37 


• *. cos. (E -{- S) = cos. 

v ' R-\-IV 

Having thus found E -f S, we have the angle E and also its equal 
CFB. Then (§ 69) 

A C = 2 R sin. \E\ B C = 2 R' sin. £ E. 

We have also A B — A C B C. , since A C and B C are in the 
same straight line (§ 32), or A B — 2 (72 -f- 72') sin. £ E. 



When the two radii are equal, the same formulae apply by making 
R' = R. In this case, we have 

I3P“ cos. (E -f- S) = cos. S — ^ ' ~2 p ^ > 


EF* A C = B C = 2R sin. \ E. 

Example. Given d = .42, g — 4.7, S = 1 ° 20 ', 6 = 11 , and the an¬ 
gles of the two frogs each 7°, to find A C = B C — \ AB. The 
common radius R, corresponding to F — 7°, is found (§51) to be 
593.5. Then 2 72 = 1187, h — 2 d = 10.16, and 10.16 -4- 1187 = 
.00856. Therefore, nat. cos. (E -f- S) = .99973 — .00856 = .99117 ; 
whence E S = 7° 37' 15". Subtracting S, we have E = 6 ° 17' 15" 
Next 

2 72 = 1187 3.074451 

%E = 3° 8 '37^" sin. 8.739106 

AC— 65.1 1.813557 













38 


CIRCULAR CURVES. 


C. Turnout from Curves. 

57. Problem. Given the radius R of the centre line of the mair 
track and the frog angle F, to determine the position of the frog by means 
of the chord B F {Jigs. 18 and 19), and to find the radius R 1 of the cen¬ 
tre line of the turnout. 


D 



Solution. I. "When the turnout is from the inside of the curve 
(fig. 18). Let A G and CF be the rails of the main track, AB the 
switch rail, and the arc B F the outer rail of the turnout, crossing the 
inside rail of the main track at F. Then, since the angle E FK has its 
sides perpendicular to the tangents of the two curves at F , it is equal to 
the acute angle made by the crossing rails, that is, E FK = F. Also 
EBL = S. The first step is to find the angle B KF denoted by K. 
To find this angle, we have in the triangle B FK (Tab. X. 14 ),BK-\- 
KF-. B K—KF= tnn. \ ( BFK+ FBK ): tan. £ ( BFK— FBK). 
But B K = R i g — and KF — R — ^ g. Therefore, B K - f- 
KF '«= 2 R — d, and BK — KF g — d. Moreover, BFK = 
BFE -f- EFK = BFE + F, and FBK = EBF — EB K — 
BFE — S. Therefore, BFK — FBK — F S. Lastly, B FK 
+ FBK= 180° — K. Substituting these values in the preceding 
t.roportion, we have 2 R — d:g — = tan. (90° — £ K ): tan. | (F-\- &), 







TURNOUT FROM CURVES. 


39 


»r ten. (90°-i£)= |U (F+ -- . But tan. (90° - $ K) 

— cot. \K = » 


V&T , •. tan. £ AT = 


_ g — <* _ 

(2 R — d) tan. £ (F-\- S) 


Next, to find the chord B F, we have, in the triangle B F C 
(Tab. X. 12 ), B F = * But Z? C= g — d, and BCF = 

180° —FCK= 180° — (90° — 1X) = 90° + £ iT, or sin. BCF 
= cos. £ K. Moreover, B F C = % (F S )for B FK — KF C 
+ B F <7, and FB K = K C F — BF C = KF C — B F C. There- 
fore, B FK — FB K — 2 BFC. But, as shown above, BFK — 
FBK=F+ S. Therefore, 2BFC= F+S,orBFC=%(F+S). 
Substituting these values in the expression for B F, we have 


BF = 


(g — d) cos. £ K 
sin. £ (F + S) 


l bf 

Lastly, to find R\ we have (§ 68 ) R 1 -f- %g — E F — -r - ^ b"ef 
But BEF — BLF — EBL, and BLF — LFK+ LKF = 
F + K. Therefore, BE F = F + K — S, and 


6 ^ R' + l9 = 


\BF 

sin. ?{F-\- K — S) 


II. When the turnout is from the outside of the curve, the preceding 
solution requires a few modifications. In the present case, the angle 
EFK* — F (fig. 19) and EB L — S. To find iT, we have in the 
triangle B F K, KF + B K : ICF — BK = tan. £ {FBK + 
BFK) : tan. J {FBK — BFK). But KF — R + | g, and BK 
— R — £ g + d. Therefore, KF -f B K = 2 R + d, and KF — 
BK — g — d. Moreover, FBK— 180° — FB L — 180° —■ 
(EBF—EBL) — 180° — (E B F — £), and BFK = 180° — 
BFK ' = 180° — (BFE + EFK’) = 180° — (EBF + F): 
Therefore, FB K — B FK = F + S. Lastly, FB K + B FK = 
180° — IC Substituting these values in the preceding proportion, we 
have 2R d : g — d — tan. (90° — % K) : tan. £ {F -f- S ), or 


tan. (90° - £ K) = (2fi+rf) g ta °^ (F +?! . But tan. (90° - £ K) 
cot. ^ K = tan ^ K ; 


. • tan. \ K = 


g — d 


(2 R -f- d) tan. ( F -f* S) 











40 


CIRCULAR CURVES. 


Next to find B F, we have, in the triangle B F 


B C sin. B C F 
sin. B F C 


But B C = g — d , and B CF = 90 c 


3 F - 
<e*r 



ein. B CF = cos. ^ K. Moreover, BFC=^(F-\-S)', for BF R 
= KFC—BFC ; and FZ? iT= 7^(7^+ BFC=KFC-\-BFC. 
Therefore, jFZJ K — BFK = 2BF C. But, as shown above, FBK — 
= Therefore, 2 2?F(7= F + S,ov BFC= ± (F + S). 

Substituting these values in the expression for B F, we have, as before. 


I3P 


BF — (9 ~ d ) cos. 

sin. £ (F + S) 


Lastly, to find R we have (§68) R f + \g — E F = 


jBF 

sin. % B EF 


by making cos. £ K = 1. This gives B F = 


* Since £ K is generally very small, an approximate vain 2 of B F may he obtained 

S — <* 

sin. £ (F + S) 

with the formula for B F in § 50. Table V. will, therefore, give a close approxima¬ 
tion to the value of B F on curves also, for any value of F contained in the table. 


, which is identical 














TURNOUT FROM CURVES. 


41 


Bat BEF — BLF — EBL, an&BLF = LFK —LKF 
F—K. Therefore, BEF — F — K — S t and 






iBF 


sin. ^ (F — K — S) 


Example. Given g = 4.7, d = .42, S = 1° 20', R = 4583.75, and 
F = 7°, to find the chord B F and the radius R' of a turnout from the 
•utside of the curve. Here 

g — d = 4.28 0.631444 0.631444 

2 R + d = 9167.92 3.962271 

£ (F + /S') = 4° 10' tan. 8.862433 sin. 8.861283 

2.824704 1.770161 

tan. 7.806740 cos. 9.999991 

1.770152 

0.301030 
sin. 8.633766 

8.934796 

R> + \g = 684.47 2.835356 

.-. R' = 682.12 


%K = 22 ' 1 . 8 " 

B F = 58.905 
2 

J (F — K — S) = 2° 27' 58.2" 


58. Problem. To find mechanically the proper position of a given 
frog. 

Solution. The method here is similar to that already given, when 
the turnout is from a straight line (§ 52). Draw B M (figs. 18 and 19) 
parallel to F C, and we have FB M — B F C — % (F S), as just 
shown (§ 57). This angle is to be laid off from B M ; but as F is the 
point to be found, the chord F C can be only estimated at first, and 
B M taken parallel to it, from which the angle ^ (F + S) may be 
laid off by the method of § 52. In this case, however, the first meas¬ 
ure on the arc is d, and not 2 d j since we have here to start from B M, 
and not from the rail. Having thus determined the point F approxi¬ 
mately, B M may be laid off more accurately, and F found anew. 

59. When frogs are cast to be kept on hand, it is desirable to have 
them of such a pattern that they will fall at the beginning or end of a 
certain rail; that is, the chord BF is known, and the angle F is re¬ 
quired. 









*2 


CIRCULAR CURVES. 


Problem. Given the position of a frog by means of the chord B F 
(Jigs. 14, 18, and 19), to determine the frog angle F. 

Solution. The formula B F — B in ' £ (F~+ ~ S ) > w ^ c k * s exact 011 
straight lines (§ 50), and near enough on ordinary curves (§ 57, note), 
gives 

By this formula ^ (F -f- S) may be found, and consequently F. 

60. Problem. Given the radius R of the centre line of the main 
track , and the radius R 1 of the centre line of a turnout, to find the frog 
angle F, and the chord B F (figs. 18 and 19). 

Solution. I. When the turnout is from the inside of the curve 
(fig. 18). In the triangle BEKfind the angle B E K and the side E K. 
For this purpose we have BE — R 1 % g, BK = R % g — d , and 
the included angle EB K = S. Then in the triangle EFK we have 
E X, as just found, E F = R' -J- ? <7, and FK = R — \g. The frog 
angle EFK — F may, therefore, be found by formula 15, Tab. X., 
which gives 

ET tan. bF= J ( s b ) if _=Zfl , 

v s (s — a) 

where s is the half sum of the three sides, a the side E K ’ and b and c 
the remaining sides. 

Find also in the triangle EFK the angle FEK , and we have the 
angle BE F — BEK — FEK. Then in the triangle BE F we 
have (§69) 

B F & 2 (R> + l g) sin. \ BE F .* 

II. When the turnout is from the outside of the curve (fig. 19). In 
the triangle BEK find the angle BEK and the side E K. For this 
purpose we have B E = R 1 -J- ^ g, B K = R — \ g -f- d, and the in¬ 
cluded angle EBK— 180° — S._ Then in the triangle EFK we 
have E K , as just found, E F = R' + \ g, and FK =R-\- % g. The 
angle EFK may, therefore, be found by formula 15, Tab. X., which 

gives tan. \ E FK — s {s — d)~^ • But an S^ e BFK' = F 


* The value of B F may be more easily found by the approximate formula B F = 

rr . J 

, and generally vdth sufficient accuracy. See note to § 57. This re¬ 


sin. 4 (F+ S) 

mark applies also to B Fin the second part of this solution. 











TURNOUT FROM CURVES. 


43 


*= 180° — E FK. Therefore ^ F = 90° — \E FK , and cot. £ F =» 
tan. ^E FIC) 

ESP . •. cot. | F = \J i.l~ b \. ( s ~ c ), 

^ s (s — a) 

where s is the half sum of the three sides, a the side E K , and b and c 
the remaining sides. 

Find also in the triangle E FK the angle FE IC, and we have the angle 
13 E F = FE K — BEK. Then in the triangle B E F we have (§69) 

I3T BF= 2 (Ri + $g) sin. \BEF. 


Example. Given g = 4.7, <7 = .42, S = 1° 20', R = 4583.75, and 
R' = 682.12, to find F and the chord B F of a turnout from the outside 
of the curve. Here in the triangle BEK (fig. 19) we have BE — 
Ri -{- l g = 684.47, B K — R — l g -f- d = 4581.82, and the angles 
BEK+ BICE = S = 1° 20'. Then 

BK— BE = 3897.35 3.590769 


& [BEK+ BKE) = 40' 

BIC+ BE = 5266.29 
l {BEIC— B ICE)* = 29.6029' 


tan. 8.065806 
T.656575 

3.721505 
tan. 7.935070 


BE IC= 1° 9.6029' 

_ __. „ , , , . I? X sin. E B K 

E K is now found by the formula A K = — elnBE K — ’ or & R 

= log. 4581.82 + log. sin. 178° 40' — log. sin. 1° 9.6029' = 3.721491, 
whence E IC — 5266.12. 

Then to find F , we have, in the triangle E FK, s = J (5266.12 -f* 
634.47 + 4586.10) = 5268.34, s — a = 2.22, s — b = 4583.87, and 
s -■ c = 682.24. 


s — b = 4583.87 
s — c = 682.24 


s = 5268.34 
s — a = 2.22 


1^=3° 30' 
-,F= 7° 


3.661233 

2.833937 

6.495170 

3.721674 

0.346353 


4.068027 
2)2^427143 
cot. 1.213571 


* This angle and the sine of 1° 9 6029' below, are found by the method given in 
•onnection with Table XIII. If the ordinary interpolations had been used, we 
should have found F = 7° 7', whereas it should be 7°, since this example is the 
•onverse of that in § 57. 











44 


CIRCULAR CURVES. 


To find FEK, we have s as before, but as a is here the side FR 
opposite the angle sought, we have s — a = 682.24, s — b = 4583.87, 
and s — c — 2.22. Then by means of the logarithms just used, we 
find \ FE K — 3° 2' 45". Subtracting \BEK — 34' 48", we have 
\BEF = 2° 27' 57". Lastly, BF = 1368.94 sin. 2° 27' 57" = 
58.897. 

The formula BF = (§ note) would give BF — 

58.906, and this value is even nearer the truth than that just found, 
owing, however, to no error in the formulae, but to inaccuracies inci¬ 
dent to the calculation. 

61. If the turnout is to reverse, in order to join a track parallel to 
the main track, as A CB (fig. 20), it will be necessary to determine 
the reversing points C and B. These points will be determined, if we 
find the angles AEG and B F C, and the chords A C and CB. 

62. Problem. Given the radius D K = R (Jig 20) of the centre 
line of the main track, the common radius EC— C F — IV of the centre 
line of a turnout , and the distance B G — b between the centre lines of the 
parallel tracks, to find the central angles AE Cand BFCand the chords 


A C and B C. 


T' 





Solution. In the triangle A E K find the angle AEK and the side 





CROSSINGS ON CURVES. 


45 


E K For this purpose we have A E = R\ A K = R — d, and the 
included angle E AK— S. Or, if the frog angle has been previously 
calculated by § 60, the values of A E K and E K are already known.* 

Find in the triangle EFK the angles E FK and FE K. For this 
purpose we have E K, as just found, E F = 2 li\ and FK = R + 
R' — h. Then AEC = AEK— FEK, and BFC = EFK . 
Lastly, (§69) 

XW* A C= 2R'sln±AE C y CB = 2 R' sin. ±B F C. 

This solution, with a few obvious modifications, will apply, when 
the turnout is from the outside of a curve. 

D. Crossings on Curves . 

63. When a turnout enters a parallel main track by a second switch, 
tt becomes a crossing. Then if the tangent points A and B (fig. 21) 
are fixed, the distance A B must be measured, and also the angles 
which A B makes with the tangents at A and B. The common ra¬ 
dius of the crossing may then be found by § 40; or if one radius of the' 
crossing is given, the other may be found by § 38. But if one tangent 
point A is fixed, and the common radius of the crossing is given, it 
will be necessary to determine the reversing point C and the tangent 
point B. These points will be determined, if we find the angles AEC 
and BFC ’, and the chords A C and CB. 

64. Problem. Given the radius D K — R {Jig. 21) of the centre 
line of the main track , the common radius EC— C F = R' of the centre 
line of a crossing , and the distance D G = b between the centre lines of the 
■parallel tracks , to find the central angles AEC and BFC and the chords 
A Cand CB. 

Solution. In the triangle AEK find the angle A E K and the side 
E K. For this purpose we have A E = R\ A K — R — d, and the 
included angle EAK— S. 

Find in the triangle B FK the angle B FK and the side FK. For 
this purpose we have BF — RBK = R — b + d, and the included 
angle FBK— 180° — S. 

Find in the triangle EFK the angles FEK and EFK. For this 


* The triangle AUK does not correspond precisely with B E K in § 60, A being 
on the centre line and B on the outer rail; but the difference is too slight to affect 
the calculations. 



46 


CIRCULAR CURVES. 


purpose we have E K and FK as just found, and E F =* 2 R’. Then 
AEC = AEK—FEK, and BFC — EFK— BFK. Lastly 
(* 69,) 

EF" AC = 2Ri sin. \AE C\ CB = 2 R< sin. %BF C. 


D 



Article IY. — Miscellaneous Problems. 

65. Problem. Given AB — a (Jig. 22) and the perpendicular 
B C — b, to Jind the radius of a curve that shall pass through C and the 
tangent point A. 

Solution. Let 0 be the centre of the curve, and draw the radii A 0 
and C 0 and the line CD parallel to A B. Then in the right triangle 
COD we have 0 C 2 = CD 2 + 0D 2 . But 0 C = R, CD — a, and 
0 D = A 0 — AD =.R — b. Therefore, R 2 = a 2 + (R — b) 2 = 
a 2 -J- E 2 — 2 R b 4- b 2 , or 2 R b = a 2 + b 2 ; 

BP .-.R=*!* 

2 b 

Example. Given a = 204 and b = 24, to find R. Here R =• 





MISCELLANEOUS PROBLEMS. 


47 


66. Corollary S. If R and b are given to find AB = a, that 
is, to determine the tangent point from which a curve of given radius 



must start to pass through a given point, we have (§65) 2Rb — 
a 5 + h 2 , or a 2 = 2 Rb — b 2 ; 

.'.a = Jb (2 R — b). 

Example. Given b — 24 and R = 879, to find a. Here a =» 
v/24 (1758 — 24) = */ 41616 = 204. 

67. Corollary 2. If R and a are given, and b is required, we 
have (§ 65) 2 Rb = a 2 + & 2 , or b 2 — 2Rb = — a*. Solving this 
equation, we find for the value of b here required, 

^ b = R — JR 2 — a 2 . 

68. Problem. Given the distance A C — c (Jig. 22) and the an¬ 
gle BA C — A, to Jind the radius R or dejlection angle D of a curve, that 
thall pass through C and the tangent point A. 

Solution. Draw 0 E perpendicular to A C. Then the angle AOE 
= \A 0 C — BA C — A (§ 2, III.), and the right triangle A OE gives 

(Tab. X.9) AO ^ * 

B" .-.R±= A L t- 

sin. A 

To find D, we have (§9) sin. D — ^. Substituting for R its value 
: ust found, we have sin. Z) = 50 -f- ° A ; 

7 pin. jl 












48 


CIRCULAR CURVES. 


. sin. D = 


100 sin. A 
c 


Example. Given c = 285.4 and A — 5°, to find R and D. Here 
142.7 „„„„ , . _ 100 sin. 50 sin. 5° . ... 

« = SETT? = 1637 ' 3 ; and “»• D = -2854- = w = Sln - 1 45 

or D = 1 ° 45'. 

69. Problem. Given the radius R or the deflection angle D of a 
curve, and the angle B A C = A [fig. 22), made by any chord with the 
tangent at A, to find the length of the chord A C — c. 

be 

Solution. If R is given, we have (§ 68) R = j ; 

BP . •. c = 2 R sin. A. 

. . . . „ „ . . ~ 100 sin. A 

If D is given, we have (§ 68) sin. D — --— ; 

100 sin. A 


sin. D 

This formula is useful for finding the length of chords, when a curve 
is laid out by points two, three, or more stations apart. Thus, suppose 
that the curve A C is four stations long, and that we wish to find the 
length of the chord A C. In this case the angle A — 4 D and c = 

1 00 sm. i D ' this method Table II. is calculated. 

6111 . 1) J 

X ■’ f 

Example. Given R = 2455.7 or D = 1 ° 10 ', and A — 4° 40', to 

find c. Here, by the first formula, c = 4911.4 sin. 4° 40' = 399.59. 

„ . , . . 100 sin. 4° 40' 

IJy the second formula, c = - jn ^5 = 399.59. 

70. Problem. Given the angle of intersection K CB = 1 [fig. 23), 
and the distance CD = b from the intersection point to the curve in • the 
direction of the centre , to find the tangent A C = T, and the radius A O 
= R. 

Solution. In the triangle ADC we have sin. CA D : sin. AD C — 
CD : AC. But CAD = b A O D = \ I 2, HI. and VI.), and as 

the sine of an angle is the same as the sine of its supplement, 

sin. AD C = sin. ADE = cos. DAE = cos. £/. Moreover, CD 
— b and AC— T. Substituting these values in the preceding pro- 

^ COS 4- jr 

portion, we have sin. \ I : cos. \ I = b : T, or T — VV 5 whence 


(Tab. X. 33) 


sin. £ I 









MISCELLANEOUS PROBLEMS. 


49 


S3P* T = b cot. £ I. 

To find R , we have (§ 5) R = T cot. ^ Z. Substituting for T its 
value just found, we have 

CF* R — b cot. \ 1 cot. £1 



bxample. Given Z = 30°, b — 130, to find Tand R. Here 


b = 130 

2.113943 

>w- 

i-h 

II 

O 

CO 

© 

cot. 0.880571 

T = 987.45 

2.994514 

JZ= 15° 

cot. 0.571948 

72 = 3685.21 

3.566462 „ 


71. Problem. Given the angle of intersection KCB — I {fig. 23), 

and the tangent A C = T, or the radius A 0 — R, to find CD — b. 

Solution. If T is given, we have (§ 70) T = b cot. £/, or b =* 
T 

aot $ I ’ 

10P . •. b = T tan. \ I. 

If R is given, we have (§ 70) R — b cot. \ I cot. |Z, or 6 =» 
R 

«Ot i Jcot. £ I ’ 


. •. 6 = R tan. £ I tan. ^ Z. 









50 


CIRCULAR CURVES. 


Example. Given I— 27°, T — 600 or R = 2499.18, to find b 
Here b = 600 tan. 6° 45' = 71.01, or b = 2499.18 tan. 6° 45 
tan. 13° 30' =71.01. 

72. Problem. Given the angle of intersection I of two tangents 
A C and B C [fig. 24), to find the tangent point A of a curve , that shedl 
pass through a point E, given by CD — a, D E = b, and the angle CD E 



Solution. Produce DE to the curve at G, and draw C 0 to the cen¬ 
tre 0. Denote DFbyc. Then in the right triangle CDF we hare 
(Tab. X. 11) DF= CD cos. CDF.\ or 

53P c = a cos. ^I. 

Denote the distance A D from D to the tangent point by x. Then, by 
Geometry, x 2 — D E X D G. But D G = D F + F G = D F + 
E F = 2 D F — D E = 2c — b. Therefore, x 2 = b (2 c — 6), and 

13 s3 x = Jb (2 c — b ). 

Having thus found A D , we have the tangent AC — AD D C 
= i + a. Hence, R or D may be found (4 5 or § 11). 

If the point E is given by EH and CH perpendicular to each other, 
a and b may be found from these lines. Por a = CH -f- D H =» 

CH + EH cot. (Tab. X. 9). and b = DE = 







MISCELLANEOUS PROBLEMS. 


51 


Example. Given I = 20° 16% a — 600, and b — 80, to find x and 
R. Here c = 600 cos. 10° 8' = 590.64, 2 c—b = 1101.28, and x = 

x/80 X 1101.28 = 296.82. Then T = 600 + 296.82 = 896.82, and 
R = 896.82 cot. 10° 8' = 5017.82. 

73. Problem. Given the tangent A C (Jig. 25), and the chord 
A B, uniting the tangent points A and B, to Jind the radius A O = R. 



Solution. Measure or calculate the perpendicular CD. Then if CD 
be produced to the centre 0, the right triangles ADC and C A O, 
having tli3 angle at C common, are similar, and give CD : AD == 
A C: A O, or 




R = 


AD X AC 
CD 


If it is inconvenient to measure the chord A B, a line E F, parallel 
to it, may be obtained by laying off from C equal distances CE and 
CF. Then measuring E G and G C\ we have, from the similar tri¬ 
angles E G C and CAO, C G: GE = AC: AO, or R= ~£* C . 

Example. Given A C = 246 and AD — 240, to find R. Here 
CD = 54, and R = = 1093.33. 









52 


CIRCULAR CURVES. 


74. Problem. Given the radius AO — R {Jig. 25), to Jind th* 
tangent A G — T of a curve to unite two straight lines given on the ground . 

Solution. Lay off from the intersection C of the given straight lines any 
equal distances CE and C F. Draio the perpendicular C G to the mid¬ 
dle cf E F, and measure G E and C G. Then the right triangles 
E G Cand C A 0, having the angle at C common, are similar, and 
give GE : C G . ~ A 0 : A C, or 

U£-p. _ C G X A 0 ' 

G E 

By this problem and the preceding one, the radius or tangent points 
of a curve may be found without an instrument for measuring angles. 


Example. Given R = 1093|, G E = 80, and C G = 18, to find T. 
IS x 1093£ 

Here 1 =- —- — 246. 


75. Problem. To find the angle of intersection l of two straight 
lines , when the point of intersection is inaccessible , and to determine the tan¬ 
gent points , when the length of the tangents is given. 

Solution. I. To find the angle of intersection I. Let A C and C V 
(fig. 26) be the given lines. Sight from some point A on one line to a 
point B on the other , and measure the angles CAB and TBV. These 
angles make up the change of direction in passing from one tangent to 
the other. But the angle of intersection (§ 2) shows the change of di¬ 
rection between two tangents, and it must, therefore, be equal to the 
sum of C A B and TBV, that is, 


I — C A B -{■ TB V 


But if obstacles of any kind render it necessary to pass from A C to 
B Fby a broken line, as AT) E F B, measure the angles C AD, ND E y 
PE F, RFB, and SB F, observing to note those angles as minus which 
are laid off contrary to the general direction of these angles. Thus the 
general direction of the angles in this case is to the right; but the 
angle PEF lies to the left of BE produced, and is therefore to be 
marked minus. The angles to be measured show the successive changes 
of direction in passing from one tangent to the other. Thus CAD 
shows the change of direction between the first tangent and A D , 
ND E shows the change between A D produced and D E, P E F the 
change between D E produced and E F,R FB the change between 
EF produced and FB, and, lastly, SB V the change between BF pro- 




MISCELLANEOUS PROBLEMS. 


58 


duced and the second tangent. But the angle of intersection (§ 2) 
shows the change of direction in passing from one tangent to another, 
and it must, therefore, be equal to the sum of the partial changes 
measured, that is, 

BP 1= CAD + NDE — PEF+ RFB + SB V. 



II. To determine the tangent points. This will be done if we find 
the distances A C and B C; for then any other distances from C may 
be found. It is supposed that the distance A B , or the distances A D, 
DE, E F, and FB have been measured. 

If one line A B connects A and B, find A C and B C in the triangle 
ABC . Tor this purpose we have one side A B and all the angles. 

If a broken line A D E F B connects A and B, let fall a 'perpendicular 
B G from B upon A C , produced if necessary , and find A G and B G 
by the usual method of working a traverse. Thus, if A C is taken as a 
meridian line, and D K, E L , and FM are drawn parallel to A C, and 
D H, E K, and FL are drawn parallel to B G , the difference of lati¬ 
tude A G is equal to the sum of the partial differences of latitude A H. 
DK, EL , and EM, and the departure B G is equal to the sum of the 
partial departures DII, E K, FL, and B M. To find these partial 
differences of latitude and departures, we have the distances A D, DE, 
E F, and F B, and the bearings may be obtained from the angles 
already measured. Thus the bearing of A D is CAD, the bearing of 
DE is KDE = KDN+ NDE = CAD-\- NDE, the bearing 
of E F is LEE = LEE — P E F KDE — PEE, and the 


54 


CIRCULAR CURVES. 


bearing of FB is M FB = M FR + R FB = LEF-\-RFB] that 
is, the bearing of each line is equal to the algebraic sum of the preced 
ing bearing and its own change of direction. The differences of lati¬ 
tude and the departures may now be obtained from a traverse table, 
or more correctly by the formulae: 

Diff. of lat. = dist. X cos. of bearing ; dep. = dist. X sin. of bearing 

Thus, AH = AD cos. C A 7>, and D H = AD sin. CA D. 

Having found A G and B G, we have, in the right triangle B G (7, 

(Tab. X. 9) GC = BG cot. B CG, and B C = faf&c G * But 
B C G= 180° —7. Therefore, cot. BCG — — cot. 7, and sin. BCG 
— sin. 7. Hence G C = ■*— B G cot. 7, and B C = * Then, 

since A'C — A G -f- G C, we have 

EF AC=AG— BG cot. 7; BC = . 

sin. 7 

When 7 is between 90° and 180°, as in the figure, cot. 7 is negative, 
and — B G cot. 7 is, therefore, positive. When I is less than 90°, G 
will fall on the other side of 7; but the same formula for A C wil still 
apply ; for cot. 7 is now positive, and consequently, — B G ‘cot. / is 
negative, as it should be, since, in this case, A C would equal A G mi 
nus G C. 

Example. Given AD— 1200, D E = 350, E F — 300, F B — 
310, CAD = 20°, NDE = 44°, PE F = — 25°, R FB = 31°. 
and SB V = 30°, to find the angle of intersection 7, and the distance? 
A C and B C. 

Here 7 = 20° + 44° — 25° + 31° -f- 30° = 100°. To find A Q 
and B Cr, the work may be arranged as in the following table: — 


Angles to 
the Right. 

Bearings. 

Distances. 

N. 

E. 

o 

20 

N. 20 E. 

1200 

1127.63 

410.42 

44 

64 

350 

153.43 

314.58 

—25 

39 

300 

233.14 

188.80 

31 

70 

310 

106.03 

291.3Q 




1620.23 

1205.10 


The first column contains the observed angles. The second contains 
the bearings, which are found from tne angles of the first column, in 
















MISCELLANEOUS PROBLEMS. 


55 


the manner already explained. A C is considered as running north 
from A , and the bearings are, therefore, marked N. E. The other col¬ 
umns require no explanation. We find A G = 1620.23, and B G = 
1205.10. Then G C — — BG cot. 1 = — 1205.1 X cot. 100° = 
212.49. This value is positive, because it is the product of two nega¬ 
tive factors, cot. 100° being the same as —cot. 80°, a negative quanti¬ 
ty. Then A C = A G + GC= 1620.23 + 212.49 = 1832.72, and 

B C = = 1223.69. Having thus found the distances of A 

and B from the point of intersection, we can easily fix the tangent 
points for tangents of any given length. 

76. Pro'blcsn* To lay out a curve , when an obstruction of any kind 
prevents the use of the ordinary methods. 



Solution. First Method. Suppose the instrument to be placed at 
A (fig. 27), and that a house, for instance, covers the station at H, and 
also obstructs the view from A to the stations at D and E. Lay off 
from A C, the tangent at A , such a mu^iple of the deflection angle D, 
| as will be sufficient to make the sight clear the obstruction. In the 
! figure it is supposed that 4 D is the proper angle. The sight will then 
pass through F, the fourth station from A, and this station will be de- 
i termined by measuring from A the length of the chord A F, found by 




56 


CIRCULAR CURVES. 


§ 69 or by Table II. From the station at .Fthe stations at D and E 
may afterwards be fixed, by laying off the proper deflections from the 
tangent at F. 

Second Method. This consists in running an auxiliary curve paral 
lei to the true curve, either inside or outside of it. For this purpose 
lay off perpendicular to A C, the tangent at A , a line A A 1 of any con 
venient length, and from A' a line A' C’ parallel to A C. Then A’ C * 
is the tangent from which the auxiliary curve A' E 1 is to be laid off. 
The stations on this curve are made to correspond to stations of 100 
feet on the true curve, that is, a radius through B' passes through B , a 
radius through D' passes through Z), &c. The chord A 1 B 1 is, there¬ 
fore, parallel to A B, and the angle C' A' B' = CA B ; that is, the de¬ 
flection angle of the auxiliary curve is equal to that of the true curve 
It remains to find the length of the auxiliary chords A 1 B\ B' D', &c 
Call the distance A A' = h. Then the similar triangles ABO and 
A' B' 0 give A 0 : A'O = A B : A'B ', or R : R — h = 100 : A' B'. 

r-M r At o, 100(72 — 6) 100 6 , 

i herefore, A' B' — — —^ -= 100 — . If the auxiliary curve 

were on the outside of the true curve, we should find in the same way 

A' B' = 100 -f- . It is well to make h an aliquot part of 11 ; for 

the auxiliary chord is then more easily found. Thus, if n is any 

whole number, and we make b = ^ , we have A 1 B 1 = 100 ± 

100 R 

— 100 ± — • If, for example, b = jqq , we have n — 100, and A' B 

— 100 ± 1 = 101 or 99. When the auxiliary curve has been run, 
the corresponding stations on the true curve are found, by laying off 
in the proper direction the distances B B', DD\ &c., each equal to b. 

77. ProMeiai. Having run a curve A B [Jig. 28), to change the 
tangent point from A to C ,, in such a wag that a curve of the same radius 
may strike a given point D. 

Solution. Measure the distance B D from the curve to I) in a direction 
parallel to the tangent CE. This direction may be sometimes judged 
of by tbe eye, or found by the compass. A still more accurate way is 
to make the angle D B E equal to the intersection angle at E , or to 
twice B AE, the total deflection angle from A to B \ or if A can be 
seen from B , the angle DBA may be made equal to B AE. 

Measure on the tangent (backward or forward , as the case may be) a dis 
tance A C = B D, and C will be the new tangent point required. For. if 
CZZbe drawn equal and parallel to A F , we have FI1 equal and par 



MISCELLANEOUS PROBLEMS. 


57 


ullel to A C, and therefore equal and parallel to B D. Hence D H — 
B F = A F = CH, and D H being equal to CH, a curve of radius 
d E from the tangent point C must pass through D. 



78, Problem. Having run a curve A B (Jig. 29) of radius B or 
deflection angle Z), terminating in a tangent B D , to find the radius R 1 or 
deflection angle D’ of a curve A C, that shall terminate in a given parallel 
tangent CE. 



Solution. Since the radii B F and C G are perpendicular to the par¬ 
allel tangents CE and B D, they are parallel, and the angle A G C = 
A FB. Therefore, A C G , the half-supplement of A G C, is equal to 
4 









58 CIRCULAR CURVES. 

ABF, the half-supplement of A FB. Hence A B and B C are in the 
same straight line, and the new tangent point C is the intersection of 
A B produced with CE. 

Represent A B by c, and AC — c B C by cK Measure B C, or, if 
more convenient , measure D C and Jind B C by calculation. To calculate 

D C 

B C from D C, we have BC= ~ in (Tab. X. 9), and the angle 
BBC — ABK—BAK, the total deflection from A to B. Then 
the triangles AFB and AG C give AB:AC=BF: C G, or c : c 1 
= R:R'- } 

R 1 = - R. 

* c 

To find D ', we have ($ 10) R' = , and R — si ^ jj • Sub- 

60 

stituting these values in the equation for R', we have - in jy =* 
c* 50 

t X 8 in. j) J 

c 

. •. sin. B' =-j sin. D. 


79. Problem. Given the length of two equal chords A C and B 0 
{Jig. 30), and the perpendicular CD, to Jind the radius R of the curve. 



Solution. From 0, the centre of the curve, draw the perpendicular 
OE. Then the similar triangles 0 BE and BCD give B 0 : BE 
~BC-.CD,ovR:%BC=BC:CD. Hence 




R = 


BC 2 
2 CD * 










MISCELLANEOUS PROBLEMS. 


59 


This problem serves to find the radius of a curve on a track already 
laid. For if from any point C on the curve we measure two equal 
chords A C and B C \ and also the perpendicular CD from C upon the 
whole chord A B, we have the data of this problem. 

80. Problem. To draw a tangent F G (Jig. 30) to a given curve 
from a given point F. 

Solution. On any straight line FA., which cuts the curve in two points , 
measure F C and FA, the distances to the curve. Then, by Geometry, 

E3P F G = JF C X FA. 

This length being measured from F, will give the point G. When 
F G exceeds the length of the chain, the direction in which to measure 
it, so that it will just touch the curve, may be found by one or two trials. 

81. Problem. Having found the radius AO — R of a curve 
(Jig. 31), to substitute for it two radii A E = R x and D F — R 21 the 
longer of which A E or BE 1 is to be used for a certain distance only at 
tath end of the curve. 



Solution. Assume the longer radius of any length which may be thought 




60 


CIRCULAR CURVES. 


proper, and Jind (§ 9) the corresponding deflection angle D t . Suppose 
that each of the curves A D and B D l is 100 feet long. Then drawing 
G0 , we have, in the triangle FOE , OE:FE — sin. OFE : sin. FOE. 
But the side OE = AE — AO — R\ — R, FE —BE — DF = 
*R x — R% j the angle FOE = 180° — AO C — 180° — £ 7, and the 
angle 0FE — A OF — 0EF = \I — 2 Z> 15 since 0 E F = 2 7), 
(§7). Substituting these values, and recollecting that sin. (180° — ^ 1) 
= sin. \ 7, we have R x — R : R x — R 2 = sin. (£7 — 2 7),) : sin. £ 7 
Hence 

R —R — ~ ^ ^ n ‘ 2 I 

1 * sin. (17 —2 A) 

77, is then easily found, and this will be the radius from D to D\ or 
until the central angle DFD' = 7 — 4 D x . 

The object of this problem is to furnish a method of flattening the 
extremities of a sharp curve. It is not necessary that the first curve 
should be ju'st 100 feet long ; in a long curve it may be longer, and in 
a short curve shorter. The value of the angle at E will of course 
change with the length of A 7), and this angle must take the place of 
2 D x in the formula. The longer the first curve is made, the shorter 
the second radius will be. It must also be borne in mind, in choosing 
the first radius, that the longer the first radius is taken, the shorter will 
be the second radius. 

Example. Given R = 1146.28 and 7 = 45°, to find R 2 , if R x is as¬ 
sumed = 1910.08, and A D and B D r each 100. Here, by Table I., 
D x = 1° 30'. Then 

R x — R = 763.8 2.882980 

£ 7 = 22° 30' sin. 9.582840 


2.465820 

£7 — 2 D x — 19° 30' sin. 9.523495 
Ri — R 2 = 875.64 2.942325 

74 = R x — 875.64 = 1034.44 


82. Problem. To locate the second branch of a compound or re • 
versed curve from a station on the first branch. 

Solution. Let A B (fig. 32) be the first branch of a compound curve, 
and D its deflection angle, and let it be required to locate the second 
branch AB\ whose deflection angle is D\ from some station B 
on A B. 





MISCELLANEOUS PROBLEMS. 


61 


Let n be the number of stations from A to B, and n' the number of sta¬ 
tions from A to any station B' on the second branch. Represent by V the 
angle A B B', which it is necessary to lay off from the chord B A to strike 
B 1 . Let the corresponding angle A B' B on the other curve be repre¬ 



sented by V. Then we have V -f V — 180° — BAB'. But if 
T T' be the common tangent at A, we have TA B + T' A B' — nD 
4 . n'D' = 180° — BAB'. Therefore, V + V = nD + n'D'. 
Next in the triangle A BB' we have sin. V ': sin. V — AB : AB f . 
But A B : A B' = n : n', nearly , and sin. V 1 : sin. V — V : V, near- 

ly. Therefore we have approximately V 1 : V = n : n', or V' = V. 
Substituting this value of V in the equation for V + V, we have 
V + jp V—nD-\-n' D'. Therefore, n' Vn V= n' (nD -{- n'D'), or 
y _ n' [n D -f- n' D') 


n 4 * n' 

The same reasoning will apply to reversed curves, the only change 
being that in this case V V 1 = n D — n ' D', and consequently 




Y _ n' {nD — n' D') 

n 4 - n' 


When in this formula n'D' becomes greater than nD, V becomes 
minus, which signifies that the angle V is to be laid off above B A in¬ 
stead of below. 

This problem is particularly useful, when the tangent point of a 
curve is so situated, that the instrument cannot be set o\er it. The 
same method is applicable, when the curve A B' starts from a straight 
line ; for then we may consider A B' as the second branch of a com¬ 
pound curve, of which the straight line is the first branch, having its 
radius equal to infinity, and its deflection angle D = 0 . Making 
D = 0 , the formula for V becomes 





62 


CIRCULAR CURVES. 


F _ n>*D> 
n -f- n 1 

When n and n r are each 1, the formula for Fis in all cases exact, 
for then the supposition that V': V= n : n' is strictly true, since A B 
will equal A B', and Fand V\ being angles at the base of an isosceles 
triangle, will also be equal. Making n and n 1 equal to 1, we have 

F=£(Z) + Z)'). 

When the curve starts from a straight line, this formula becomes, by 
making D — 0, 

F= 

We have seen that when n or n' is more than 1, the value of F is 
only approximate. It is, however, so near the truth, that when nei¬ 
ther n nor n' exceeds 3, the error in curves up to 5° or 6° varies from 
a fraction of a second to less than half a minute. The exact value of 
F might of course be obtained by solving the triangle A B B\ in 
which the sides A B and AB' may be found from Table II., and the 
included angle at A is known. The extent to which these formulae 
may be safely used may be seen by the following table, which gives 
the approximate values of Ffor several different values of n, n\ Z), 
aud D\ and also the error in each case. 


Compound Curves. 


n. 

D. 

n ( . 

D'. 

V. 

Error. 


o 


o 

o » 

it 

1 

0 

5 

1 

4 10 

0.9 

1 

0 

5 

3 

12 30 

25.3 

2 

0 

3 

3 

5 24 

22.1 

3 

0 

3 

3 

4 30 

29.7 

1 

1 

5 

3 

13 20 

18.6 

2 

h 

1 

3 

1 20 

0.7 

2 

2 

3 

3 

7 48 

15.0 

2 

2 

4 

3 

10 40 

24.7 

3 

3 

3 

4 

10 30 

54.0 


Reversed Curves. 

n. 

D. 

»'. 

D'. 

V. 

Error. 

i 

o 

3 

4 

o 

3 

O 1 

7 12 

n 

27.2 

2 

3 

4 

3 

4 0 

23.5 

3 

3 

4 

3 

1 42® 

8.3 

3 

2 

3 

3 

3 45 

24.0 

2 

1 

1 

4 

0 40 

0.1 

2 

1 

4 

2 

4 0 

11.0 

1 

6 

2 

6 

4 0 

23.5 

1 

5 

3 

5 

7 30 

51.8 

2 

3 

5 

3 

6 25f 

52.8 


As the given quantities are here arranged, the approximate values 
of Fare all too great; but if the columns n and n’ and the columns D 
and D r were interchanged, and Fcalculated, the approximate values 
of F would be just as much too small, the column of errors remaining 
the same. 






























MISCELLANEOUS PROBLEMS. 


63 


83. Problem. To measure the distance across a river on a given 



Solution. First Method. Let A B (fig. 33) be the required distance. 
Measure a line A C along the bank, and take the angles B AC and 
ACB. Then in the triangle A B C we have one side and two angles 
to find A B. 

If A C is of such a length that an angle A CB — \DA C can be 
laid off to a point on the farther side, we have ABC=\DAC=* 
ACB. Therefore, without calculation, A B — A C. 



Second Method. Lay off A C (fig. 34) perpendicular to A B. Meas¬ 
ure A C , and at C lay off CD perpendicular to the direction CB , and 
meeting the line of A B in D. Measure A D. Then the triangles 
A CD and ABC are similar, and give AD \ AC — AC \ AB. 

Therefore, AB — . 

If from (7, determined as before, the angle A CB' be laid off* equal 
to A CB, we have, without calculation, A B = A B r . 

Third Method. Measure a line AD (fig. 35) in an oblique direction 
from the bank, and fix its middle point C. From any convenient 
point E in the line of A B , measure the distance E C, and produce 
















64 


MISCELLANEOUS PROBLEMS. 


E C until CF —EC. Then, since the triangles ACE and DCF 
are similar by construction, we see that D F is parallel to E B. Find 



now a point G, that shall be at the same time in the line of CB and 
of D F, and measure G D. Then the triangles ABC and D G C are 
equal, and GD is equal to the required distance A B. 

As the object of drawing EF is to obtain a line parallel to A B, this 
line may be dispensed with, if by any other means a line G F be drawn 
through D parallel to A B. A point G being found on this parallel in 
the line of CB } we have, as before, G D = A B. 















PARABOLIC CURVES. 


65 


CHAPTER II. 

PARABOLIC CURVES. 

Article I. — Locating Parabolic Curves. 

84. Let AE B (fig. 36) be a parabola, A C and B C its tangents, 
and A B the chord uniting the tangent points. Bisect A B in Z>, and 
oin CD. Then, according to Analytical Geometry, — 



I. CD is a diameter of the parabola, and the curve bisects CD in E- 

II. If from any points !F, TT n , &c., on a tangent AF, lines be 
nrawn to the curve parallel to the diameter , these lines T M : T' M , 
'1 I. 11 III. IV. M", &c., called tangent deflections , will be to each other as the 
squares of the distances A T, A T\ A T n , &c. from the tangent 
point A. 

III. A line ED (fig. 37), drawn from the middle of a chord A B to 
the curve, and parallel to the diameter, may be called the middle ordi 
nate of that chord; and if the secondary chords A E and BE be drawn, 
the middle ordinates of these chords, K G and L H , are each equal to 
\ED. In like manner, if the chords A K, KE, E L, and L B be 
drawn, their middle ordinates will be equal to \ K G or \ L H. 

IV. A tangent to the curve at the extremity of a middle ordinate, 
is parallel to the chord of that ordinate. Thus MF, tangent to the 
curve at E , is parallel to A B. 




66 


PARABOLIC CURVES. 


V. If any tAvo tangents, as A C and B C, be bisected in 31 and 
the line M F ' joining the points of bisection, will be a new tangent, 
middle point E being the point of tangency. 

85. Problem. Given the tangents A C and B C, equal or unequal\ 
(Jig. 36,) and the chord A B, to lay out a parabola by tangent deflections. 


Solution. Bisect A B in D, and measure C D and the angle A CD $ 
or calculate CD * and A CD from the original data. Divide the tan¬ 
gent A C into any number n of equal parts, and call the deflection 
TM for the first point a. Then (§ 84, II.) the deflection for the sec¬ 
ond point will be T* M 1 — 4 a, for the third point T" M n = 9 a, and 
so on to the nth point or C, Avhere it will be n 2 a. But the deflection 
at this last point is CE — J CD (§ 84,1.). Therefore, n 2 a = C E, 
and 


Having thus found a, Ave have also the succeeding deflections 4 a, 9 a, 
16 a, &c. Then laying off at T , T 1 , &c. the angles A TM , A T' M\ 
&c. each equal to A C D, and measuring down the proper deflections, 
just found, the points 31, 3D, &c. of the cun ? e will be determined. 

The curve may be finished by laying off on A C produced n parts 
equal to those on A C, and the proper deflections will be, as before, a 
multiplied by the square of the number of parts from A. But an 



* Since CD is drawn to the middle of the base of the triangle ABC, we have, by 
Geometry, C D* = * (A C* +.B C 2) — A D\ 


5 ’ ** 





LOCATING PARABOLIC CURVES. 


61 


easier way generally of finding points beyond E is to divide the sec¬ 
ond tangent B C into equal parts, and proceed as in the case of A C. 
If the number of parts on B C be made the same as on A C , it is obvi¬ 
ous that the deflections from both tangents will be of the same length 
for corresponding points. The angles to he laid off from B C must, 
of course, be equal to B CD. 

The points or stations thus found, though corresponding to equal 
distances on the tangents, are not themselves equidistant. The length 
of the curve is obtained by actual measurement. 

86 . Problem. Given the tangents A C and B C, equal or unequal, 
{Jig. 37,) and the chord A B , to lay out a parabola by middle ordinates. 


C 



Solution. Bisect AB in D, draw CD, and its middle point E will 
be a point on the curve (§ 84,1.). D E is the first middle ordinate, 
and its length may be measured or calculated. To the point E draw 
the chords A E and BE, lay off the second middle ordinates G iTand 
II L, each equal to ^ D E (§ 84, III ), and K and L are points on the 
curve. Draw the chords AK, KE, EL, and LB, and lay off third 
middle ordinates, each equal to one fourth the second middle ordi¬ 
nates, and four additional points on the curve will be determined. 
Continue this process, until a sufficient number of points is obtained 

87. Problem. To draw a tangent to a parabola at any station. 

Solution . I. If the curve has been laid out by tangent deflections 
(§ 85), let M ,n (fig. 36) be the station, at which the tangent is to be 
drawn. From the preceding or succeeding station, lay off, parallel to 
CD, a distance M n N or EL equal to a, the first tangent deflection 
(§ 85), and M ni N or M'" L will be the required tangent. The same 
Ihing may be done by laying off from the second station a distance 
M' T 1 = 4a, or at the third station a distance GP — 9a; for the 





68 


PARABOLIC CURVES. 


required tangent will then pass through T' or G. It will be seen, 
also, that the tangent at ill m passes through a point on the tangent at 
A corresponding to half the number of stations from A to M ,n ; that 
is, M m is four stations from A, and the tangent passes through T\ 
the second point on the tangent A C. In like manner, M m is six sta¬ 
tions from B, and the tangent passes through G , the third point on the 
tangent B C. 

II. If the curve has been laid out by middle ordinates (§ 86), the tan¬ 
gent deflection for one station is equal to the last middle ordinate made 
use of in laying out the curve. For if the tangent A C (fig. 37) were 
divided into four equal parts corresponding to the number of stations 
from A to E , the method of tangent deflections would give the same 
points on the curve, as were obtained by the method of § 86. In this 
case, the tangent deflection for one station would he a = is CE = 
izDE] but the last middle ordinate was made equal to \ GK or 
K D E. Therefore, a is equal to the last middle ordinate, and a tan¬ 
gent may be drawn at any station by the first method of this section. 

A tangent may also be drawn at the extremity of any middle ordi¬ 
nate, by drawing a line through this extremity, parallel to the chord 
of that ordinate (§ 84, IY.). 

88. In laying out a parabola by the method in § 85, it may some¬ 
times be impossible or inconvenient to lay off all the points from the 
original tangents. A new tangent may then be drawn by § 87 to any 
station already found, as at M ,n (fig. 36), and the tangent deflections 
a, 4 a, 9 a, &c. may be laid off from this tangent, precisely as from the 
first tangent. These deflections must be parallel to CD, and the dis¬ 
tances on the new tangent must be equal to. T*N or NM'", which 
may be measured. 

89. Problem. Given the tangents A C and B C, equal or unequal, 
[fig 38,) to lay out a parabola by bisecting tangents. 

Solution. Bisect A C and B C in D and F, join D F, and find E , the 
middle point of D F. E will be a point on the curve (§ 84, V.). We 
have now two pairs of what may be called second tangents, A D and 
D E, and E F and FB. Bisect A D in G and D E in H, join G H, 
and its middle point M will be a point on the curve. Bisect E F and 
FB in K and L, join KL, and its middle point N will be a point on 
the curve. We have now four pairs of third tangents, A G and G M, 
MH and HE, EK and KN, and NL and L B. Bisect each pair in 
turn, join the points of bisection, and the middle points of the joining 


LOCATING PARABOLIC CURVES. 69 


iines will be four new points, M’, M", N", and N r . The same method 
may be continued, until a sufficient number of points is obtained. 



90. ProbleiM. Given the tangents A C and 13 Coequal or unequal^ 
< fy' ^9,) and the chord A 13 , to lay out a parabola by intersections. 



Solution. Bisect A B in D, draw CD, and. bisect it in E. Divide 
ihe tangents A C and B C , the half-chords A D and D B, and the line 
CE, into the same number of equal parts ; five, for example. Then 
the intersection M of A a and F G will be a point on the curve. For 
FM = 5 Ca, and Ca — \ GE. Therefore, FM = 55 CE , which is 
the proper deflection from the tangent ati^to the curve (§ 85). In 
like manner, the intersection N of A b and HK may be shown to be a 
point on the curve, and the same is true of all the similar intersections 
indicated in the figure. 

If the line D E were also divided into five equal parts, the line A a 
would be intersected in Mon the curve by a finedrawn from 23 through 
a', the fine A b would be intersected in N on the curve by a fine drawn 








70 


PARABOLIC CURVES. 


from B through b', and in general any two lines, drawn from A and B 
through two points on CD equally distant from the extremities C and 
D , will intersect on the curve. To show this for any point, as M, it is 
sufficient to show, that B a’ produced cuts F G on the curve; for it 
has already been proved, that A a cuts F G on the curve. Now 
Da' :M G — BD:B G = 5 : 9, orMG=§Da'. But £>a' = ] C£.. 
Therefore, MG = h C E. Again, FG: CD = AG: AD = 1:5. 
Therefore, FG = lCD = lCE. We have then FM = FG — 
MG = f CE — 2 ? CE = §5 CE. As this is the proper deflection 
from the tangent at F to the curve (§ 85), the intersection of B a' with 
F G is on the curve. This furnishes another method of laying out a 
parabola by intersections. 

91. The following example is given in illustration of several of the 
preceding methods. 

Example. Given A C — B C = 832 (fig. 40), and AB = 1536, to 
lay out a parabola A E B. We here find CD = 320. To begin with 
the method by tangent deflections (§ 85), divide the tangent A C into 

C E 160 

eight equal parts. Then a = = 2.5. Lay off from the 

divisions on the tangent Fl = 2.5, G 2 =*4 X 2.5 = 10, H 8 = 

9 X 2.5 = 22.5, and A 4 = 16 X 2.5 = 40. Suppose now that it is 
inconvenient to continue this method beyond IC In this case we may 


C 



find a new tangent at E , by bisecting A C and B C 89), and draw¬ 
ing KL through the points of bisection. Divide the new tangent 
KE = | A D — 384 into four equal parts, and lay off from KE the 





RADIUS OF CURVATURE. 


7 ] 


same tangent deflections as were laid off from A K , namely, M 5 = 
22.5, iV6 = 10, and 07 = 2.5. To lay off the second half of the 
curve by middle ordinates (§ 86), measure EB = 784.49. Bisect 
E B in P, and lay off the middle ordinate P R = £ D E — 40. 
Measure E R = 386.08, and BR = 402.31, and lay off the middle or¬ 
dinates S T and VW, each equal to £P R — 10. By measuring the 
chords E T, TR, R W, and WB , and laying off an ordinate from 
each, equal to 2.5, four additional points might be found. 


Article II. — Radius of Curvature. 

92. The curvature of circular arcs is always the same for the same 
arc, and in different arcs varies inversely as the radii of the arcs. 
Thus, the curvature of an arc of 1,000 feet radius is double that of an 
arc of 2,000 feet radius. The curvature of a parabola is continually 
changing. In fig. 39, for example, it is least at the tangent point A , 
the extremity of the longest tangent, and increases by a fixed law, un¬ 
til it becomes greatest at a point, called the vertex, where a tangent to 
the curve would be perpendicular to the diameter. From this point 
to B it decreases again by the same law. We may, therefore, con¬ 
sider a parabola to be made up of a succession of infinitely small cir¬ 
cular arcs, the radii of which-continually increase in going from the 
vertex to the extremities. The radius of the circular arc, correspond¬ 
ing to any part of a parabola, is called the radius of curvature at that 
point. 

If a parabola forms part of the line of a railroad, it will be necessa¬ 
ry, in order that the rails may be properly curved (§ 28), to know 
how the radius of curvature may be found. It will, in general, be 
necessary to find the radius of curvature at a few points only. In 
short curves it may be found at the two tangent points and at the mid¬ 
dle station, and in longer curves at two or more intermediate points 
besides. The rails curved according to the radius at any point should 
be sufficient in number to reach, on each side of that point, half-way to 
the next point. v 

93 . Problem. To find the radius of curvature at certain stations 
on a parabola. 

Solution. Let AEB (fig. 41) be any parabola, and let it be re¬ 
quired to find the radii of curvature at a certain number of stations 


72 


PARABOLIC CURVES. 


from A to E. These stations must be selected at regular interval 
from those determined by any of the preceding methods. Let n de 
note the number of parts into which A E is divided, and divide CL 
into the same number of equal parts. Draw lines from A to the points 



of division. Thus, if n == 4, as in the figure, divide CD into four 
equal parts, and draw A F } A E, and A G. Let A D = c, A F = c l 
A E = c 2 , A G — c 3 , and A C — T. Denote, moreover, C D by d 
and the area of the triangle A CB by A. Then the respective radii 
for the points E, 1,2, 3, and A will be 





C 3 

*3 = i, 



The area A may be found by form. 18, Tab. X.; c and T are known ; 
and c x , c 2 , c 3 may be found approximately by measurement on a figure 
carefully constructed, or exactly by these general formulae : — 


C’j 2 = c 2 + 


-a 2 = cA 

= c 3 2 
&c. 


T 2 — c 2 (u —l)d 2 


+ 


n 

— 

n 2 

+ 

2^2 

— 

c 2 

(n — 3) tP 


n 


n 2 

1 


— 

c 2 

(;n — 5) d 2 

-r 


n 


n 2 

+ 


— 

c 2 

(n — 7) d 2 


&c. 


It will be seen, that each of these values is formed from the preceding, 
by adding the same quantity T n C -, and subtracting ^ multiplied in 
Succession by h, l, r. - 2.s 5, &cr Making n » 4* wc have 











RADIUS OF CURVATURE. 


73 


«. 2 = » ! +i{T*-<?)-&<P, 

«2 S = C! 5 + 1 (T J — C ! ) — ,V rf 8 , 

«.*■■■«»* + i (r s -c») +A<P. 

All the quantities, which enter int? the expressions for the radii, are 
now known, and the radii may, therefore, he determined. The same 
method will apply to the other half of the parabola. 

The manner of obtaining the preceding formulae is as follows. The 
radius of curvature at any given point on a parabola is, by the Differ- 

j) 

ential Calculus, R — which p represents the parameter of 

the parabola for rectangular coordinates, and E the angle made with 
a diameter by a tangent to the curve at the given point. First, let the 
middle station E (fig. 42) be the given point. Then the angle E is the 



angle made with ED by a tangent at E , or since A B is parallel to 
the tangent at E (§ 84, IV.), sin. E — sin. A DE — sin. B DE. Let 
p' be the parameter for the diameter E D. Then, by Analytical Ge 

ometry, p = p' sin. 2 E. Therefore, at this point R = 2 gi ^ 3 E = 


p'sin.2 E p' AD 2 

2 sin. 3 E 2 sin. E ’ ■ But P ~ ED 


c2 

id' 


Therefore, R = 


= c dsL ' E = 1 5 since A = c d sin * E ( Tab - X * 17 )- 

Next, to find R l , or the radius of curvature at E , the first station 
from E. Through H draw F G parallel to CD, and from F'draw the 
tangent FK. Join A K , cutting CD in L. Then from what has just 
been proved for the radius of curvature at E , we have for the radius 
A G 3 

of curvature at H, R x — ^ p % • Now A G : A L — AF \ A C = 









74 


PARABOLIC CURVES. 


n — 1 : n, or A G = X AL. But AL = c 1 For, since A F — 
X A C, the tangent deflection FH — ^ (§ 84,11.), and 

F G = 2 FH= (j ^~d. Then, since CL : FG = A C: A F = 

n:n — 1 , CL — X F G = d. Hence L D = d — ^ <* 
= \d, that is, J.L = Ci . Substituting this value in the expres¬ 


sion for A G above, we have A G = 


AF = 


n — 1 


- c x . Moreover, since 
n - X AC, add because similar triangles are to each other a? 
the squares of their homologous sides, we have the triangle AF G — 
X A CL. But A CL : A CD == CL:CD = n-l:n, or 
A CL = X A CD. Therefore, AFG = (n ~, 1)3 X ^ CD, and 
A FK = 2 A FG = x A CB = A. Substituting 


A G3 

these values of A G and A FK in the equation R x = A F ^ > and re¬ 
ducing, we find R x — . By similar reasoning we should find R 2 = 

«2 3 7? C 3 3 P „ 

X» — A » &c - 

It remains to find the values of Cj, c 2 , &c. Through A draw A M 
perpendicular to CD, produced if necessary. Then, by Geometry, we 
have A D 8 = AL 2 -f- L D 2 — 2 L D X L M, and AC 2 = A L 2 
CL 2 + 2 CL X LM. Finding from each of these equations the 
value of 2 L M, and putting these values equal to each other, we have 
AIA + Lm-Am ACl-A V-CV ' But = 


LD 


CL 


AD = c, AC = T, and CL = d. Substituting these value* 
in the last equation, and reducing, we find 


ji2 

*’ = - + 


• l)c* (n—1 )rf* 
; — „* 


By similar reasoning we should find 



2 T 2 

, (» 

— 2)c 2 

2(n — 2)e?* 

c 2 3 = 

n 

+ 

n 

n* 


3T* 

(n 

— 3)c* 

3(n — 3)d» 

c 3 — 

n 

T 

n 

n 2 

&C. 



&c. 

























RADIUS OF CURVATURE. 


75 


From these equations the values of c x 2 , c 2 2 , c 3 2 , &c. given on page 72 
are readily obtained. That given for is obtained from the first of 
these equations by a simple reduction ; that given for c 2 2 is obtained 
by subtracting the first of these equations from the second, and reduc¬ 
ing ; that given for c 3 2 is obtained by subtracting the second equation 
from the third, and reducing; and so on. 

94. Example. Given (fig. 41) A C = T = 600, B C = T' = 520, 
and AD — c — 550, to find R, R x , R 2 , R 3 , and R A , the radii of cur¬ 
vature at E, 1, 2, 3, and A. 

To find CD = d, we have, by Geometry, d 2 = ^ ( T 2 + T' 2 ) — c 2 
which gives d 2 = 12700. 

To find the area of A CB = A, we have (Tab. X. 18) A — 
y/s (s — a) (s — b) (s — c). 


s = 1110 
t — a 590 
s — 6 = 510 
s — c = 10 


log. A 


3.045323 

2.770852 

2.707570 

1.000000 

2)9.523745 

4.761872 


Next i (r* - c*) i (T+ c) (T-c) = 60 = 14375, and 

</s 12700 

— 793.75. Then 

c 2 = 550 2 = 302500 

c 2 = 302500 + 14375 — 3 X 793.75 = 314493.75 
Co 2 = 314493.75 + 14375 — 793.75 = 328075 
c 3 2 = 328075 + 14375 + 793.75 = 343243.75 
c 3 

To find R , w r e have R = , or log. R — 3 log. c — log. A. 

c = 550 2.740363 

n3 


& 

A 

R = 2878.8 


8.221089 

4.761872 

3.459217 


To find R x , we have R x = , or log?Ri =-|-log. c x 2 — log. A. 


c x * = 314493.75 

ci 3 

A 

R x = 3051.7 


5.49761 
8.246418 
4.76 872 


3.484546 










76 


PARABOLIC CURVES, 


In the same way we should find R 2 — 3251.5, R 3 = 3479.6, R 4 =» 
3737.5. 

To find the radii for the second part EB of the parabola, the same 
formulas apply, except that T' takes the place of T. We have then 

\(T'* - c«) = i(T' + c) (T 1 - c) = 1070 ^ ~ 30 = — 8025 
Hence 


c x * = 302500 — 8025 — 2381.25 = 292093.75 
c 2 2 = 292093.75 — 8025 — 793.75 = 283275. 
c 3 2 = 283275.— 8025 + 793.75 = 276043.75 


C 3 g 

To find R x , we have R x = ~ , or log. R x — -z log. c x 2 — log. A 


292093.75 


5.465523 


; e 


8.198284 0 


A 

R x == 2731.6 


4.761872 


3.436412 


In the same way we should find R 2 = 2608.8, R 3 = 2509.5, R 4 * 
2433. 


It will be seen, that the radii in this example decrease from one ta 
gent point to the other, which shows that both tangent points lie <1 
the same side of the vertex of the parabola ($ 92). This will be t 
case, whenever the angle B CD, adjacent to the shorter tangent, e ; 
ceeds 90°, that is„ whenever c 2 exceeds T' 2 + d 2 . If B CD = 90 : 
the tangent point B falls on the vertex. If B CD is less than 90 
one tangent point falls on each side of the vertex, and the curvatu ■ 
will, therefore, decrease towards both extremities. 

95. If the tangents T and T' are equal, the equations for c,* c 2 2 , & 
will be more simple; for in this case d is perpendicular to c, and 1 
— c 2 = cP. Substituting this value, we get 


<F 

*_ r 1 r — 

1 _c + n 2 » 


Example. 


c 4 ’ = c,’ + -^-, 

5 (P 

C 3 2 = c 2 2 + ' n 2 * 

&C. &C. 


Given, as in § 91, T ~ T> — 832, c = 768, and d 






RADIUS OF CURVATURE. 


77 


320, to find the radii 72, 72 x , and 72, at the points E, 4, and A (fig. 40) 
Here A = cd = 245760, n = 2, and c z 2 = c 2 + = 615424 

c3 c2 7682 c ,3 ^ T3 

d 320 ~ 1843 . 2 , R l — — , and 7i, = "Td ’ 


Then 72 


A 


c x 2 = 615424 
Ci 3 

cd = 245760 
7?! = 1964.5 

T = 832 

2^3 

cc? = 245760 
72, = 2343.5 


5.789174 

8.683761 

5.390511 

3.293250 

2.920123 

8.760369 

5.390511 

3.369858 


is the radius at the point 72 also, and 72 s the radius at the point B 














78 


LEVELLING. 


CHAPTER III, 


LEVELLING. 


Article I. — Heights and Slope Stakes. 


96. The Level is an instrument consisting essentially of a telescope, 
supported on a tripod of convenient height, and capable of being so 
adjusted, that its line of sight shall be horizontal, and that the tel¬ 
escope itself may be turned in any direction on a vertical axis. The 
instrument when so adjusted is said to be set. 

The line of sight, being a line of indefinite length, may be made to 
describe a horizontal plane of indefinite extent, called the plane of tht 
level. 

The levelling rod is used for measuring the vertical distance of any 
point, on which it may be placed, below the plane of the level. Thil 
distance is called the sight on that point. 

A* 3 

97. Problem* To find the difference of level of two points, as A 


xi 


and B {fig. 43). 

Solution. Set the level between the two points,* and take sights on 
both points. Subtract the less of these sights from the greater, and' 
the difference will be the difference of level required. For if F P rep-p¬ 
resent the plane of the level, and A G be drawn through A parallel to| 
FP, A F will be the sight on A , and B P the sight on B. Then the-'- 
required difference of level B G — B P — P G = B P — A F. 

If the distance between the points, or tue nature of the ground,? 1 
makes it necessary to set the level more than once, set down all the { 
backward sights in one column and all the forward sights in another. \ 
Add up these columns, and take the less of the two sums from the 
greater, and the difference will be the difference of level required. 
Thus, to find the difference of level between A and D (fig. 43), the 
level is first set between A and B , and sights are taken on A and B ; 
the level is then set between B and C, and sights are taken on B and 


* The level should be placed midway between the two points, when practicable, 
in order to neutralize the effect of inaccuracy in the adjustment of the instrument, 
and for the reason given in § 105. 



HEIGHTS AND SLOPE STAKES, 


79 


C; lastly, the level is set between C and D , and sights are taken 

on C and D. Then the difference of 
level between A and D is E D = 
(. BP+KC+ OD)—{AF+BI-\- 
NC ). For E D = HC — L C — 
HM + M C— L C. But HM.= B G 
= BP — A F, MC=KC — BI, 
and L C = N G — OD. Substituting 
these values, we have E D — B P — 
AF+KC — BI — NC + OD = 
(BP + KC + OD) — {AF-\- BI 
+ NC). 

98. It is often convenient to refer all 
heights to an imaginary level plane 
called the datum plane. This plane 
may be assumed at starting to pass 
through, or at some fixed distance above 
or below, any permanent object, called 
a bench-mark , or simply a bench. It is 
most convenient, in order to avoid mi¬ 
nus heights, to assume the datum plane 
at such a distance below the bench¬ 
mark, that it will pass below all the 
points on the line to be levelled. Thus 
if A B (fig. 44) were part of the line to 
be levelled, and if A were the starting 
point, we should assume the datum 
plane CD at such a distance below 
some permanent object near A , as 
would make it pass below all the points 
on the line. If, for instance, we had 
reason to believe that no point on this 
line was more than 15 or 20 feet below 
A, we might safely assume CD to be 
25 feet below the bench near A, in 
which case all the distances from the 
line to the datum plane would be posi¬ 
tive. Lines before being levelled are 
usually divided into regular stations, the height of each of which above 
the datum plane is required. 
























Fig. 44. 


80 


LEVELLING. 


R 


99. Problem. To fnd the heights above a datum plane of the sev 

eral stations on a given line. 

Solution. Let A B (fig. 44) represent 
a portion of the line, divided into regu 
lar stations, marked 0, 1,2, 3, 4, 5, &c 
and let CD represent the datum plane, 
assumed to be 25 feet below a bench¬ 
mark near A. Suppose the level to he 
set first between stations 2 and 3, and a 
sight upon the bench-mark to be taken, 
and found to be 3.125. Now as this 
sight shows that the plane of the level 
E F is 3.125 feet above the bench-mark 
and as the datum plane is 25 feet be¬ 
low this mark, we shall find the height 
of the plane of the level above the da¬ 
tum plane by adding these heights, 
which gives for the height of E F 25 -f- 
3.125 = 28.125 feet. This height may 
for brevity’s sake be called the height 
of the instrument , meaning by this the 
height of the line of sight of the instru¬ 
ment. 

If now a sight be taken on station 0, 
we shall obtain the height of this sta¬ 
tion above the datum plane, by sub¬ 
tracting this sight from the height of 
the instrument; for the height of this 
station is 0 C and 0 C — E C — E 0. 
Thus if E0 = 3.413, 0 C — 28.125 — 
3.413 = 24.712. In like manner, the 
heights of stations 1,2, 3, 4, and 5 may 
be found, by taking sights on them in 
succession, and subtracting these sights 
from the height of the instrument. 
Suppose these sights to be respective¬ 
ly 3.102, 3.827, 4.816, 6.952, and 9.016, 
and we have 

height of station 0 = 28.125 — 3.413 = 24.712, 

“ “ « 1 = 28.125 — 3.102 = 25.023, 


l 


00 


/t>% 




■5 


W 4 

























HEIGHTS AND SLOPE STAKES. 


81 


height of station 2 = 28.125 — 3.827 '= 24.298, 
“ “ “ 3 = 28.125— 4.816 = 23.309, 

f{ “ “ 4 = 28.125 — 6.952 = 21.173, 

“ “ “ 5 — 28.125 — 9.016 = 19.109. 


Next, set the level between stations 7 and 8, and as the height of sta¬ 
tion 5 is known, take a sight upon this point. This sight, being added 
to the height of station 5, will give the height of the instrument in its 
new position ; fo'r G K = G 5 4- 5 K. Suppose this sight to be G 5 
= 2.740, and we have GK= 19.109 + 2.740 = 21.849. A point 
like station 5, which is used to get the height of the instrument after 
resetting, is called a turning point. The height of the instrument being 
found, sights are taken on stations 6, 7, 8, 9, and 10, and the heights 
of these stations found by subtracting these sights from the height of 
the instrument. Suppose these sights to be respectively 3.311, 4.027, 
3.824, 2.516, and 0.314, and we have 

height of station 6 = 21.849 — 3.311 = 18.538, 

“ “ « 7 = 21.849 — 4.027 = 17.822, 

“ “ “ 8 = 21.849 — 3.824 = 18.025, 

“ “ “ 9 = 21.849 — 2.516 = 19.333, 

“ “ “ 10 = 21.849 — 0.314 = 21.535. 

The instrument is now again carried forward and reset, station 1C 
is used as a turning point to find the height of the instrument, and 
every thing proceeds as before. 

At convenient distances along the. line, permanent objects are se 
lected, and their heights obtained and preserved, to be used as starting 
points in any further operations. These are also called benches. Let 
us suppose, that a bench has been thus selected near station 9, and 
that the sight upon it from the instrument, when set between stations 
7 and 8, is 2.635. Then the height of this bench will be 21.849 — 
2.635 = 19.214. 

100. From what has been shown above, it appears that the first 
thing to be done, after setting the level, is to take a sight upon some 
point of known height, and that this sight is always to be added to the 
known height, in order to get the height of the instrument. This first 
sight may therefore be called a plus sight. The next thing to be done 
is to take sights on those points whose heights are required, and to 
subtract these sights from the height of the instrument, in order to get 
the required heights. These last sights may therefore be called minus 
sights 


82 


LEVELLING. 


101. The field notes are kept in the following form. The first col 
umn in the table contains the stations, and also the benches marked 
B., and the turning points marked t.p., except when coincident with 
a station. The second column contains the plus sights ; the third col¬ 
umn shows the height of the instrument ; the fourth contains the minus 
sights; and the fifth contains the heights of the points in the first column. 


Station 

+s. 

II. I. 

— s. 

H. 

B. 

3.125 



25.000 

0 


28.125 

3.413 

24.712 

1 



3.102 

25.023 

2 



3.827 

24.298 

3 



4.816 

23.309 

4 



6.952 

21.173 

5 

2.740 


9.016 

19.109 

6 


21.849 

3.311 

18.538 

7 



4.027 

17.822 

8 



3.824 

18.025 

9 



2.516 

19.333 

B. 



2.635 

19.214 

10 



0.314 

21.535 


The height of the bench is set down as assumed above, namely, 25 
feet; the first plus sight is set opposite B., on which point it was 
taken, and, being added to the height in the same line, gives the height 
of the instrument, which is set opposite 0; the minus sights are set 
opposite the points on which they are taken, and, being subtracted 
from the height of the instrument, give the heights of these points, as 
set down in the fifth column. The minus sights are subtracted from 
the same height of the instrument, as far as the turning point at station 
5, inclusive. The plus sight on station 5 is set opposite this station, 
and a new height obtained for the instrument by adding the plus sight 
to the height of the turning point. This new height of the instrument 
is set opposite station 6, where the minus sights to be subtracted from 
it commence. These sights are again' set opposite the points on which 
they were taken, and, being subtracted from the new height of the in¬ 
strument, give the heights in the last column. 


102. Problem. To set slope stakes for excavations and embank- 
inents. 

Solution. Let A BIIK C (fig. 45) be a cross-section of a proposed 
excavation, and let the centre cut A M = c, and the width of the road- 












HEIGHTS AND SLOPE STAKES. 


83 


bfid tlK — b. The slope of the sides B H or C K is usually given by 
the ratio of the base KN to the height E N. Suppose, in the present 
case, that KN : E N = 3 : 2, and we have the slope = §. Then if 
the ground were level, as D A E, it is evident that the distance from 


Fig. 45. 


c 





the centre A to the slope stakes at D and E would be A D = A E = 
MK-\-KN=^b + | c. But as the ground rises from A to C 
through a height C G = g, the slope stake must be set farther out a 
distance E G = %g\ and as the ground falls from A to B through a 
height B F = g, the slope stake must be set farther in a distance D F 



To find B and C, set the level, if possible, in a convenient position 
for sighting on the points A, B , and C. From the known cut at the 
centre find the value of A E = ^b lie- Estimate by the eye the 
rise from the centre to where the slope stake is to be set, and take thi3 
as the probable value of g. To A E add § g, as thus estimated, and 
measure from the centre a distance out, equal to the sum. Obtain 
now by the level the rise from the centre to this point, and if it agrees 
with the estimated rise, the distance out is correct. But if the esti¬ 
mated rise prove too great or too small, assume a new value for g, 
measure a corresponding distance out, and test the accuracy of the 
estimate by the level, as before. These trials must be continued, until 
the estimated rise agrees sufficiently well with the rise found by the 
level at the corresponding distance out. The distance out will then be 
£6 + |c + §sr. The same course is to be pursued, when the ground 
falls from the centre, as at B ; but as g here becomes minus, the dis¬ 
tance out, when the true value of g is found, will be AF — AD — 


DF= %b A-lc — lg. 


For embankment, the process of setting slope stakes is the same as 
for excavation, except that a rise in the ground from the centre on 
embankments corresponds to a fall on excavations, and vice versd. 
This will be evident by inverting figure 45, which will then represent 








84 


LEVELLING. 


an embankment. What was before a fall to B, becomes now a rise, 
and what was before a rise to C, becomes now a fall. 

"When the section is partly in excavation and partly in embankment, 
the method above applies directly only to the side which is in excava 
tion at the same time that the centre of the road-bed is in excavation, 
or in embankment at the same time that the centre is in embank¬ 
ment. On the opposite side, however, it is only necessary to make c 
in the expressions above minus , because its effect here is to diminish 
the distance out. The formula for this distance out will, therefore, be¬ 
come %b — i c + § < 7 . 


Article II. — Correction for the Earth’s Curvature and 
for Refraction. 

103. Let AC (fig. 46) represent a portion of the earth’s surface. 
Then, if a level be set at A, the line of sight of the level will be the tan¬ 
gent A D, while the true level will be A C. The difference D C be¬ 
tween the line of sight and the true level is the correction for the 
earth’s curvature for the distance AD. 

104. A correction in the opposite direction arises from refraction. 
Refraction is the change of direction which light undergoes in passing 
from one medium into another of different density. As the atmos¬ 
phere increases in density the nearer it lies to the earth’s surface, light, 
passing from a point B to a lower point A, enters continually air of 
greater and greater density, and its path is in consequence a curve 
concave towards the earth. Near the earth’s surface this path may be 
taken as the arc of a circle whose radius is seven times the radius of 
the earth* Now a level at A, having its line of sight in the direction 
A D , tangent to the curve A J5, is in the proper position to receive the 
light from an object at B; so that this object appears to the observer 
to be at D. The effect of refraction, therefore, is to make an object 
appear higher than its true position. Then, since the correction for 
the earth’s curvature D C and the correction for refraction D B are in 
opposite directions, the correction for both will be B C = D C —D B. 


* Peirce’s Spherical Astronomy, Chap. X., § 125. It should he observed, how¬ 
ever, that the effect of refraction is very uncertain, varying with the state of the 
atmosphere. Sometimes the path of a ray is even made convex towards the earth, 
and sometimes the rays are refracted horizontally as well as vertically. 



earth’s curvature and refraction. 


P5 

This correction must be added to the height of any object as deter¬ 
mined by the level. 

105. Pl'Ol)lcm. Given the distance AD = D (Jig. 46), the radim 
of the earth AE = R, and the radius of the arc of refracted light = 7 li, 
*o find the correction B C = d for the earth's curvature and for refraction. 


A D 



Solution. To find the correction for the earth’s curvature D C, we 
have, by Geometry, DC(DC+2EC)=A Z) 2 , or D C (D (7-f 2 R) 
= D 2 . But as D C is always very small compared with the diameter 
of the earth, it may be dropped from the parenthesis, and we have 

2)2 

D C X 2 R = D 2 , or D C = %% . The correction for refraction DB 

may be found by the method just used for finding D C , merely chang- 

2)2 

ing R into 7 R. Hence D B = . We have then d — B C = 

2)2 2)2 

D C — D B = 2 r — , or 


d = 


3D 2 
7 R * 


By this formula Table III. is calculated, taking R = 20,911,790 ft., 
as given by Bowditch. The necessity for this correction may be 
avoided, whenever it is possible to set the level midway between the 
| points whose height is required. In this case, as the distance on 
each side of the level is the same, the corrections will be equal, and 
will destroy each other. 





86 


LEVELLING. 


Article III. — Vertical Curves. 


106. Vertical curves are used to round off the angles formed by 
the meeting of two grades. Let A C and CB (fig. 47) be two grades 
meeting at C. These grades are supposed to be given by the rise per sta¬ 
tion in going in some particular direction. Thus, starting from A, the 
grades of A C and CB may be denoted respectively by g and g’ ; that 
is, g denotes what is added to the height at every station on A C, and 
g' denotes what is added to the height at every station on CB ; but 
since C B is a descending grade, the quantity added is a minus quan¬ 
tity, and g’ will therefore be negative. The parabola furnishes a very 
simple method of putting in a vertical curve. 

107. Problem. Given the grade g of A C [fig. 47), the grade g 
of C B, and the number of stations n on each side of C to the tangent points 
A and By to unite these points by a parabolic vertical curve. 



Solution. Let A E B be the required parabola. Through B and G 
draw the vertical lines F K and C H, and produce AC to meet FK 
in F. Through A draw the horizontal line A K, and join A B, cut¬ 
ting C H in D. Then, since the distance from C to A and B is meas¬ 
ured horizontally, we have AH — HK , and consequently AD —i 
DB. The vertical line CD is, therefore, a diameter of the parabola 
(§ 84, I.), and the distances of the curve in a vertical direction from 
the stations on the tangent A F are to each other as the squares of the 
number of stations from A (§ 84, II.). Thus, if a represent this dis¬ 
tance at the first station from A, the distance at the second station 
would be 4 a, at the third station 9 a , and at By which is 2 n stations 

FB 

from A, it would be 4 n 2 a ; that is, FB = 4 n 2 a, or a = . To find 

a, it will then be necessary to find FB first. Through. C draw the 
horizontal line C G and we have, from the equal triangles CFG and 









VERTICAL CURVES. 


87 


A C H, F G = CII. But C H is the rise of the first grade g in the n 
stations from A to C\ that is, GII — ng, or F G = ng. GB is also 
the rise of the second grade g' in n stations, but since g' is negative 
(§ 106), we must put GB — — ng'. Therefore, FB = F G + GB 

— ng — ng'. Substituting this value of FB in the equation for a 

. ng — ng 1 

we have a = —— , or 

a = 9—j[ 

4 n 

The value of a being thus determined, all the distances of the curve 
from the tangent AF, viz. a, 4 a, 9 a, 16 a, &c., are known. Now 
if T and I' 1 be the first and second stations on the tangent, and verti¬ 
cal lines TP and T 1 P' be drawn to the horizontal line A IC, the 
height TP of the first station above A will be g , the height T' P' of 
the second station above A will be 2 g , and in like manner for suc¬ 
ceeding stations we should find the heights 3g, 4 < 7 , &c. As we have 
already found TM = a, T' M' — 4 a, &c., we shall have for the 
heights of the curve above the level of A , MP = TP — TM — 
g — a, M' P' = T' P' — T' M f = 2 g — 4 a, and in like manner for 
the succeeding heights 3 g — 9 a, 4 <7 — 16 a, &c. Then to find the 
grades for the curve at the successive stations from A, that is, the rise 
of each height over the preceding height, we must subtract each 
height from the next following height, thus: (g — a) —0 — g — a, 
(2g — 4a)—(g — a) =g — 3 a, (3# — 9 a) — (2g — 4a) = g — 5a, 
(4 g — 16 a) — (3 g — 9 a) — g — 7 a, &c. The successive grades for 
the vertical curve are , therefore, 

IdF 3 g — a, g — 3 a, g — 5 a, g — 7 a, &c. 

In finding these grades, strict regard must be paid to the algebraic 
signs. The results are then general; though the figure represents 
but one of the six casfes that may arise from various combinations of 
ascending and descending grades. If proper figures were drawn to 
represent the remaining cases, the above solution, with due attention 
to the signs, would apply to them all, and lead to precisely the same 
formulae. 

108 . Examples. Let the number of stations on each side of G y be 3, 
and let A C ascend .9 per station, and CB descend .6 per station. Here 

g — gi . 9 — (—. 6 ) 1.5 

k = 3, g = .9, and g' = —. 6 . Then, a == = 4 x 3 ' = 12 

— .125, and the grades from A to B will be 






89 


LEVELLING. 


g — a = 3 —.125 = .775, 

g — 3 a = .9 —.375 = .525, 

g — 5 a = .9 — .625 = .275, 

g — 7 a = .9 — .875 = .025, 

g— 9a = .9 — 1.125 = — .225, 

^ — 11 a = .9 — 1.375 = — .475. 

As a second example, let the first of two grades descend .8 per sta¬ 
tion, and the second ascend .4 per station, and assume two stations on 
each side of C as the extent of the curve. Here g = — . 8 , g' = .4, 

and n = 2 . Then a = — j 8 — 2 : ~ = —jp* = — .15, and the four grades 
required will be 

g—a = — .8— (—.15)= —.8 + .15 = — .65, 

g — 3 a = ; — .8 — (—.45)= —.8 + .45 = — .35, 

g — 5 a = '— .8 — (—.75)= —.8 + .75 = — .05, 

g — 7a = — .8 — (— 1.05) = — .8 + 1.05 = + .25. 

It will be seen, that, after finding the first grade, the remaining grades 
may be found by the continual subtraction of 2 a. Thus, in the first 
example, each grade after the first is .25 less than the preceding grade, 
and in the second example, a being here negative, each grade after 
the first is .3 greater than the preceding grade. 

109. The grades calculated for the whole stations, as in the fore¬ 
going examples, are sufficient for all purposes except for laying the 
track. The grade stakes being then usually only 20 feet apart, it will 
be necessary to ascertain the proper grades on a vertical curve for 
these sub-stations. To do this, nothing more is necessary than to let g 
and g' represent the given grades for a sub-station of 20 feet, and n the 
number of sub-stations on each side of the intersection, and to apply the 
preceding formula. In the last example, for instance, the first grade 
descends .8 per station, or .16 every 20 feet, the second grade ascends 
.4 per station, or .08 every 20 feet, and the number of sub-stations in 
200 feet is 10 . We have then g = — .16, g } = .08, and n = 10 

Hence a = — ^ iq ~ = = — *00®. The ^ rst S ra< ^ e there 

fore, g — a = — .16 + .006 = — .154, and as each subsequent grade 
increases .012 (§ 108), the whole may be written down without farther 
trouble, thus: — .154, — .142, —.130, — .118, — .106, — .094, —.082, 
— .070, —.058, —.046, —.034, —.022, —.010, +.002, + .014, + . 0 ^ 
+ .038 + .050, + .062, + .074. 







ELEVATION OF THE OUTER RAIL ON CURVES. 89 


Article IV. — Elevation oe the Outer Eail on Curves. 


110. Fs’oMeiia. Given the radius of a curve R, the gauge of the 
track g, and the velocity of a car per second v, to determine the proper ele¬ 
vation e of the outer rail of the curve. 

Solution. A car moving on a curve of radius R , with a velocity per sec- 

ond = r, has, by Mechanics, a centrifugal force = . To counteract 

this force, the outer rail on a curve is raised above the level of the 
inner rail, so that the car may rest on an inclined plane. This eleva¬ 
tion must be such, that the action of gravity in forcing the car down 
the inclined,plane shall be just equal to the centrifugal force, which 
impels, it in the opposite direction. Now the action of gravity on a 
body resting on an inclined plane is equal to 32.2 multiplied by the 
ratio of the height to the length of the plane. But the height of the 
plane is the elevation e, and its length the gauge of the track g. This 
action of gravity, which is to counteract the centrifugal force, is, there- 
32 2 1 

fore, =. -f- . Putting this equal to the centrifugal force, we have 


82.2 e _ ij 2 

s “-JB* 


Hence 


e 


= g yl 

32.2 R * 


If we substitute for R its value (§ 10 ) R = D , we have e = 
^0 ~ X*32 2 = -00062112 g v 2 sin. D. If the velocity is given in miles 
per hour, represent this velocity by M, and we have v = • 

Substituting this value of v, we find e = .0013361 g M 2 sin. D. When 
g = 4.7, this becomes e — .00627966 M 2 sin. D. By this formula 
Table IV. is calculated. In determining the proper elevation in any 
given case, the usual practice is to adopt the highest customary speed 
of passenger trains as the value of M. 

111 . Still the outer rail of a curve, though elevated according to the 
preceding formula, is generally found to be much more worn than the 
inner rail. On this account some are led to distrust the formula, and 
to give an increased elevation to the rail. So far, however, as the 
centrifugal force is concerned, the formula is undoubtedly correct, and 
the evil in question must arise from other causes, — causes which are 
not counteracted by an additional elevation of the outer rail. The 
principal of these causes is probably improper “ coning” of the wheels. 
Two wheels, immovable on an axle, and of the same radius, must, if 






90 


LEVELLING. 


no slip is allowed, pass over equal spaces in a given number of revo¬ 
lutions. Now as the outer rail of a curve is longer than the inner rail, 
the outer wheel of such a pair must on a curve fall behind the inner 
wheel. The first effect of this is to bring the flange of the outer wheel 
against the rail, and to keep it there. The second is a strain on the 
axle consequent upon a slip of the wheels equal in amount to the dif 
ference in length of the two rails of the curve. To remedy this, con¬ 
ing of the wheels was introduced, by means of which the radius of the 
outer wheel is in effect increased, the nearer its flange approaches the 
rail, and this wheel is thus enabled to traverse a greater distance than 
the inner wheel. 

To find the amount of coning for a play of the wheels of one inch, 
let r and r 1 represent the proper radii of the inner and outer wheels 
respectively, when the flange of the outer wheel touches the rail. Then 
r' — r will be the coning for one inch in breadth of the tire. To ena¬ 
ble the wheels to keep pace with each other in traversing a curve, their 
radii must be proportional to the lengths of the two rails of the curve, 
or, which is the same thing, proportional to the radii of these rails. If 
R be taken as the radius of the inner rail, the radius of the outer rail 
will be R -f- g, and we shall have r : r' = R : R -f- g. Therefore, r R 
-J- rg = r f R. or 

r'-r = r J.. 

R 

As an example, let R — 600, r = 1.4, and g — 4.7. Then we have 
1.4 X 4.7 

r f — r = - gQQ — .011 ft. For a tire 3.5 in. wide, the coning 
would be 3.5 X .011 = .0385 ft., or nearly half an inch. Wheels 
coned to this amount would accommodate themselves to any curves 
of not less than 600 feet radius. On a straight line the flanges of the 
two wheels would be equally distant from the rails, making both 
wheels of the same diameter. On a curve of say 2400 feet radius, the 
flange of the outer wheel would assume a position one fourth of an 
inch nearer to the rail than the flange of the inner wheel, which would 
increase the radius of the outer wheel just one fourth of the necessary 
increase on a curve of 600 feet. Should the flange of the outer wheel 
get too near the rail, the disproportionate increase of the radius of this 
wheel would make it get the start of the inner wheel, and cause the 
flange to recede from the rail again. If the shortest radius were taken 

14x47 

as 900 feet, r and g remaining the same, we should have r f — r = -- qqq ~t 




ELEVATION OF THE OUTER RAIL ON CURVES. 91 


s= .0073, and for the coning of the whole tire 3.5 X .0073 = .0256 ft., 
or about three tenths of an inch. Wheels coned to this amount would 
accommodate themselves to any curve of not less than 900 feet radius. 
If the wheels are larger-, the coning must be greater, or if the gauge of 
the track is wider, the coning must be greater. If the play of the 
wheels is greater, the coning may be diminished. Hence it might be 
advisable to increase the play of the wheels on short curves, by a slight 
increase of the gauge of the track. 

Two distinct things, therefore, claim attention in regard to the mo¬ 
tion of cars on a curve. The first is the centrifugal force, which is 
generated in all cases, when a body is constrained to move in a cur¬ 
vilinear path, and which may be effectually counteracted for any given 
velocity by elevating the outer rail. The second is the unequal length 
of the two rails of a curve, in consequence of which two wheels fixed 
on an axle cannot traverse a curve properly, unless some provision is 
made for increasing the diameter of the outer wheel. Coning of the 
wheels seems to be the only thing yet devised for obtaining this in¬ 
crease of diameter. At present, however, there is little regularity 
either in the coning itself, or in the distance between the flanges of 
wheels for tracks of the same gauge. The tendency has been to di¬ 
minish the coning,* without substituting any thing in its place. If the 
wheels could be made to turn independently of each other, the whole 
difficulty would vanish ; but if this is thought to be impracticable, the 
present method ought at least to be reduced to some system. 


* Bush and Lobdell, extensive wheel-makers, say, in a note published in Apple- 
tons’ Mechanic’s Magazine for August, 1852, that wheels made by them for the New 
York and Erie road have a coning of but one sixteenth of an inch. This coning on 
a track of six feet gauge with the other data as given above, would suit no curve 
of less than a mile radius. 



92 


EARTH-WORK. 


CHAPTER IV. 

EARTH-WORK. 

Article I. — Prismoidal Formula. 

112. Earth-work includes the regular excavation and embank 
ment on the line of a road, borrow-pits, or such additional excavations 
as are made necessary when the embankment exceeds the regular ex 
cavation, and, in general, any transfers of earth that require calcula¬ 
tion. We begin with the prismoidal formula, as this formula is fre¬ 
quently used in calculating cubical contents both of earth and masonry. 

A prismoid is a solid having two parallel faces, and composed of 
prisms, wedges, and pyramids, whose common altitude is the perpen¬ 
dicular distance between the parallel faces. 

113. Problem. Given the areas of the parallel faces B and 
the middle area M, and the altitude a of a prismoid , to find its solidity S. 

Solution. The middle area of a prismoid is the area of a section 
midway between the parallel faces and parallel to them, and the alti¬ 
tude is the perpendicular distance between the parallel faces. If now 
b represents the base of any prism of altitude a , its solidity is a b. If b 
represents the base of a regular wedge or half-parallelopipedon of alti¬ 
tude a, its solidity is %ab. If b represents the base of a pyramid of 
altitude a, its solidity is J a b. The solidity of these three bodies ad 
mits of a common expression, which may be found thus. Let m rep¬ 
resent the middle area of either of these bodies, that is, the area of a 
section parallel to the base and midway between the base and top. In 
the prism, m = b, in the regular wedge, m = %b, and in the pyramid, 
m — \ b. Moreover, the upper base of the prism = 6, and the upper 
base of the wedge or pyramid = 0. Then the expressions a 6, |a5, 
and £a& may be thus transformed. Solidity of 

prism = a6 = “ X 66 = “ (6 +6 + 46) =“(6+6+4/n), 

6 6 6 

wedge = £ a 6 = - X 36 = “ (0 + 6 + 2 6) = -(0 + 6+ 4 m), 

6 6 6 

pyramid = £a6 = -X26 = “(0 + 6 + 6) = f (0 + 6 + 4 vi). • 

6 6 6 


EORROW-PITS. 


93 


Hence, the solidity of either of these bodies is found by adding togeth¬ 
er the area of the upper base, the area of the lower base, and four 
times the middle area, and multiplying the sum by one sixth of the 
altitude. Irregular wedges, or those not half-parallelopipedons, may 
be measured by the same rule, since they are tho sum or difference of 
a regular wedge and a pyramid of common altitude, and as the rule 
applies to both these bodies, it applies to their sum or difference. 

Now a prismoid, being made up of prisms, wedges, and pyramids of 
common altitude with itself, will have for its solidity the sum of the 
solidities of the combined solids. But the sum of the areas of the 
upper and lower bases of the combined solids is equal to B + B\ the 
sum of the areas of the parallel faces of the prismoid; and the sum of 
the middle areas of the combined solids is equal to M, the middle area 
of the prismoid. Therefore 

^ S = -{B + B' + 4 M). 

6 


Article II. — Borrow-Pits. 

114. For the measurement of small excavations, such as borrow- 
pits, &c., the usual method of preparing the ground is to divide the 
surface into parallelograms * or triangles, small enough to be consid¬ 
ered planes, laid off from a base line, that will remain untouched by 
the excavation. A convenient bench-mark is then selected, and levels 
taken at all the angles of the subdivisions. After the excavation is 
made, the same subdivisions are laid off from the base line upon the 
bottom of the excavation, and levels referred to the same bench-mark 
are taken at all the angles. 

This method divides the excavation into a series of vertical prisms, 
generally truncated at top and bottom. The vertical edges of these 
prisms are known, since they are the differences of the levels at the 
top and bottom of the excavation. The horizontal section of the 
prisms is also known, because the parallelograms or triangles, into 
which the surface is divided, are always measured horizontally. 

115. Problem. Given the edges h, h x , and h 2 , to find the solidity 


* If the ground is divided into rectangles, as is generally done, and one side be 
made 27 feet, or some multiple of 27 feet, the contents may he obtained at once in 
cubic yards, by merely omitting the factor 27 in the calculation. 



94 


EARTH-WORK. 


S of a vertical prism, whether truncated or not , whose horizontal section is 
a triangle of given area A. 



Solution. When the prism is not truncated, we have h — h t — ho- 
The ordinary rule for the solidity of a prism gives, therefore, S = Ah 
>= A X £ [h + h Y + h 2 ). When the prism is truncated, let A B C- 
F G H (fig. 48) represent such a prism, truncated at the top. Through 
the lowest point A of the upper face draw a horizontal plane ADB 
cutting off a pyramid, of which the base is the trapezoid B D E C, and 
the altitude a perpendicular let fall from A on D E. Represent this 
perpendicular by p, and we have (Tab. X. 52) the solidity of the pyra- 
mid = ip X BDEC = kp X DE X $ {B D + C E) = \p X 
DE X h [B D CE) = A X h ( BD + CE), since \p X DE 
=■ A D E = A. But J [B D + CE) is the mean height of the verti¬ 
cal edges of the truncated portion, the height at A being 0. Hence 
the formula already found for a prism not truncated, will apply to the 
portion above the plane A D E, as well as to that below. The same 
reasoning would apply, if the lower end also were truncated. Hence, 
for the solidity of the whole prism, whether truncated or not, we have 

S= AX 1(11 + 11!+ h 2 ). 

116. Problem. Given the edges h , h l , 7i 2 , and h 3 , to find ths 
solidity S of a vertical prism , whether truncated or not, whose horizontal 
section is a parallelogram of given area A. 






BORROW-PITS. 


95 


Solution. Let BII (fig. 49) represent such a prism, whether trun¬ 
cated or not, and let the plane B FIID divide it into two triangular 



prisms AFll and CFII. The horizontal section of each of these 
prisms will be £ A , and if h , h t , h 2 , and h 3 represent the edges to which 
they are attached in the figure, we have for their solidity ($ 115) 
A FII — £ A x £ + hi Ar lh)i an( l CFH =%A X £ + l '2 *£* 

h 3 ). Therefore, the whole prism will have for its solidity S = £ A X 
£ {h + 2 h t + /< 2 + 2 A 3 ). Let the whole prism be again divided by 
the plane AE G C into tw'o triangular prisms BEG and DEG 
Then we have for these prisms, BEG — £ A Xi(H^i + /< 2 )> 
and D E G — £ A X £ (ft A 2 + ^})j and for the whole prism, S = 
£ A X 1 (2/t -f/t, + 2/i 2 -f A 3 ). Adding the two expressions found 
for S, we have 2 S = £ A -f- -f h 2 -j- A 3 ), or 

<5 = A X £ (A -f- h x -j- hz -f- ^ 3 ). 

It will be seen by the figure, that £ (A -J- A 2 ) = ATL = £ (7/t + A 3 ), 
or h 4 - /tj = + h 3 . The expression for £ might, therefore, be re¬ 

duced to S = A X £ (7* -f- 4 2 ), or S = yt X |(/'i + 7/ 3 ). But as 
the ground surfaces A B CD and EF GH are seldom perfect planes, 
it is considered better to use the mean of the four heights, instead of 
the mean of two diagonally opposite. 

117. Corollary. When all the prisms of an excavation have 
the same horizontal section A , the calculation of any number of them 







96 


EARTH-WORK. 


may be performed by one operation. Let figure 50 be a plan of such 
an excavation, the heights at the angles being denoted by a, a x , a 2 , 


a- (i/ _ «*» 


b 

br 


l)n b -e l?5 

n 

C/ 


c* 

CJ, da 

& 

(Zl 

fix 

d* 



Fig. 50. 


6 1 ? &c. Then the solidity of the whole will be equal to \A multi¬ 
plied by the sum of the heights of the several prisms (§ 116). Into 
this sum the corner heights a, a 2 , b, b 5 , c 5 , d , and d 4 will enter but 
once , each being found in but one prism ; the heights a x , b 4 , c, d x , d 2J 
and d 3 will enter twice, each being common to two prisms; the heights 
b l} 6 3 , and c 4 will enter three times, each being common to three 
prisms; and the heights 6 2 j c u c 2 5 an d c 3 will enter four times, each 
being common to four prisms. If, therefore, the sum of the first set of 
heights is represented by s l , the sum of the second by s 2 , of the third 
by s 3 , and of the fourth by s 4 , we shall have for the solidity of all tho 
prisms 

B = 4 A. (sj -f- 2 s 2 -f- 3 -}- 4 s 4 ). 


Article III. — Excavation and Embankment. 

118. As embankments have the same general shape as excavations, 
it will be necessary to consider excavations only. The simplest case 
is when the ground is considered level on each side of the centre line. 
Figure 51 represents the mass of earth between two stations in an ex¬ 
cavation of this kind. The trapezoid GBFH is a section of the 
mass at the first station, and G x B x F x H x a section at the second sta¬ 
tion ; A E is the centre height at the first station, and A 1 E x the centre 
height at the second station ; H F x F is the road-bed, G Gi B x B the 













CENTRE HEIGHTS ALONE GIVEN. 


97 


surface of the ground, and G G X H X H and BB X F X F the planes form¬ 
ing the side slopes. This solid is a prismoid, and might be calculated 
by the prismoidal formula (§ 113). The following method gives tne 
same result. 


A. Centre Heights alone given. 

119. Problem. Given the centre heights c and c 1 , the width of the 
road-bed b , the slope of the sides s , and the length of the section l , to find 
die solidity S of the excavation. 



Solution. Let c be the centre height at A (fig. 51) and c x the height 
at A x . The slope s is the ratio of the base of the slope to its perpen¬ 
dicular height (§ 102). We have then the distance out AB = £6 + 
s c, and the distance out A x B x — ^ b + s Ci (§ 102). Divide the whole 
mass into two equal parts by a vertical plane A A x E l E drawn 
through the centre line, and let us find first the solidity of the right- 
hand half. Through B draw the planes B E E x , B Ai E\, and 
B E x Fi , dividing the half-section into three quadrangular pyramids, 
havipg for their common vertex the point B , and for their bases the 
planes A A x E x E, EE x F x F , and A x B x F x E x . For the areas of these 
bases we have 

Area of AA X E X E =\EEi X (AE~\~A X E X ) =£Z(c + c x ), 

“ “ EE X F X F ~EFxFE x =±bl, 

“ “ A 1 B,F 1 E l =.iA I E l X(E l F 1 +A l B,) = i(bc I + sc l ’) i 

and for the perpendiculars from the vertex B on these bases, produced 
when necessary, 





98 


EARTH-WORK. 


Perpendicular on AA 1 E l E — AB — % b + s c, 

“ “ EE\F l F = AE = c, 

“ “ A 1 B l F 1 E 1 = EE t = l. 

Then (Tab. X. 52) the solidities of the three pyramids are 

B - A A x E l E =i(j5 + sc) X §■ l ( c “f“ c i) =11 c + 2 ^ c i ■+ 

sc 2 4- see j) 

B - EE l F 1 F =| cX^bl =--\lbc, 

B - A 1 B 1 FiE 1 = i/ X f (&c x 4 - scf) = \l (bci -f scf). 

Their sum, or the solidity of the half-section, is 

= \l [§& (c + c x ) + s (c 2 + Ci 2 + ccJl. 


Therefore the solidity of the whole section is 

S = £ l [i b (c + cj + s (c 2 + c x 2 + c Cj)J, 
or 

S = il [b (c + c x ) 4- I s (c 2 + cf + cCi)J 

When the slope is 1 ^ to 1 , s = §, and the factor §s = l may be 
dropped. 

120 . Problem. To find the solidity S of any number n of succes¬ 
sive sections of equal length. 

Solution. Let c, c 1 , c 2 , c 3 , &c. denote the centre heights at the suc¬ 
cessive stations. Then we have (§ 119) 

Solidity of first section = [b (c 4- Ci) I s (c 2 + c \ 4" c c x )], 

“ “ second section = ^ / [6 (c x 4 - c 2 ) 4~ S s ( c i 2 + c 2 2 4" c i c 2 )], 

“ “ third section = £ / [6 (c 2 4- c s) + § s (c 2 2 c 3 2 4 - c 2 c 3 )], 

&c. &c. 


For the solidity of any number n of sections, we should have \ l mul¬ 
tiplied by the sum of the quantities in n parentheses formed as those 
iust given. The last centre height, according to the notation adopted, 
will be represented by c n , and the next to the last by c n — \. Collect¬ 
ing the terms multiplied by b into one line, the squares multiplied by 
| s into a second line, and the remaining terms into a third line, we 
have for the solidity of n sections 


sr s=$i 


b (c 4 - 2 c x 4 - 2 c 2 4 - 2 c 3 . 4 - 2 c„ _ 1 4 - c„) 

+ § * (c 2 4 - 2 Cj 2 4 - 2 c 2 2 4 - 2 c 3 2 - 4 - 2 c\ _ 1 4 - c\) 

4- I S (c c x 4- c r c 2 4- C0C3 4- c 3 C 4 . . . . 4 - Cn- 1 Cn). 


When s = 1, the factor § s = 1 may be dropped. 




CENTRE AND SIDE HEIGHTS GIVEN. 


99 


Example. Given l — 100, b — 28, s = §, and the stations and cen¬ 
tre heights as set down in the first and second columns of the annexed 
table. The calculation is thus performed. Square the heights, and 
set the squares in the third column. Form the successive products 
cc x , ^Ca, &c., and place them in the fourth column. Add up the last 
three columns. To the sum of the second column add the sum itself, 
minus the first and the last height, and to the sum of the third column 
add the sum itself, minus the first and the last square. Then 86 is the 
multiplier of b in the first line of the formula, 592 is the second line, 
since § s is here 1, and 274 is the third line. The product of 86 by b 
= 28 is 2408, and the sum of 274, 592, and 2408 is 3274. This mul¬ 
tiplied by ^ = 50 gives for the solidity 163,700 cubic feet. 


Station. 

c. 

c2. 

CCi. 

0 

2 

4 


1 

4 

16 

8 

2 

.7 

49 

28 

3 

6 

36 

42 

4 

10 

100 

60 

5 

7 

49 

70 

6 

6 

36 

42 

7 

4 

16 

24 


46 

40 

86 

28 

2408 


306 274 

286 592 

592 2408 

2)3274 
163700. 


B. Centre and Side Heights given. 

121. When greater accuracy is required than can be attained by the 
preceding method, the side heights and the distances out (§ 102) are 
introduced. Let figure 52 represent the right-hand side of an excava 
tion between two stations. AA 1 B l B is the ground surface ; A E = c 
and A x E x = c x are the centre heights ; B G = h and B t G l = h l , the 
side heights ; and d and d lt the distances out, or the horizontal distan¬ 
ces of B and B x from the centre line. The whole ground surface 
may sometimes be taken as a plane, and sometimes the part on each 
side of the centre line may be so taken ; * but neither of these suppo- 


* It is easy in any given case to ascertain whether a surface like A Ai 2>j B is a 

















100 


EARTH-WORK. 


sitions is sufficiently accurate to serve as the basis of a general method. 
In most cases, however, we may consider the surface on each side of 
the centre line to be divided into two triangular planes by a diagonal 
passing from one of the centre heights to one of the side heights. A 
ridge or depression will, in general, determine which diagonal ought 
to be taken as the dividing line, and this diagonal must be noted in 
the field. Thus, in the figure a ridge is supposed to run from B to 
A x , from which the ground slopes downward on each side to A and 
B x . Instead of this, a depression might run from A to B x , and the 
ground rise each way to A x and B. If the ridge or depression is very 
marked, and does not cross the centre or side lines at the regular sta¬ 
tions, intermediate stations must be introduced to make the triangular 
planes conform better to the nature of the ground. If the surface 
happens to be a plane, or nearly so, the diagonal may be taken in 
either direction. It will be seen, therefore, that the following method 
is applicable to all ordinary ground. When, however, the ground is 
very irregular, the method of § 127 is to be used. 

122. Problem. Given the centre heights c and c x , the side heights 
on the right h and h l , on the left h' and h' x , the distances out on the right 
d and d l , on the left d' and d' x , the width of the road-bed b, the length of the 
section l, and the direction of the diagonals , to find the solidity S of the 
excavation. 

Solution. Let figure 52 represent the right-hand side of the excava¬ 
tion, and let us suppose first, that the diagonal runs, as shown in the 
figure, from B to A x . Through B draw the planes B E E x , B A x E x , 
and B E X F X , dividing the half-section into three quadrangular pyra¬ 
mids, having for their common vertex the point B , and for their bases 
the planes A A 1 B 1 E, E E x F x F\ and A x B x F x E x . For the areas 
of these bases we have 

Areaof A A X E X E = £ EE X x [A E + A x E x ) — (c + c x ), 

“ “ EE X F X F =EFxEE 1 = 

11 11 -A-i B x F l E x = ^ A x E x X di + j B x F x X h x = £ d x c x -f- \ b h x , 

and for the perpendiculars from the vertex B on these bases, produced 
when necessary, 

plane ; for if it is a plane, the descent from A to B will be to the descent from A x to 
B i, as the distance out at the first station is to the distance out at the second sta¬ 
tion, that is, c — h : c x — hi = d : di. If we had c = 9, h = 6, e x = 12, h x = 8, 
d = 24, and di =27, the formula would give 3 : 4 = 24 : 27, which shows that the 
surface is not a plane. 



» CENTRE AND SIDE HEIGHTS GIVEN 


10 ) 


Perpendicular on A A x E l E —EG — d, 
“ “ EE J F X F =BG =h, 

“ “ A 1 B l F l E l = EE, = l. 


Fig. 52. 



Then (Tab. X. 52) the solidities of the three pyramids are 

B-AA 1 E 1 E = £ cl X 2 1 ( c + c i) = 11 (dc -f- 
B-EE l F 1 F =\lbh, 

B - A 1 B l F 1 E 1 = £1 x h( d i c ! + h hh i) = W (rfiCi+£&*,). 

Their sum, or the solidity of the half-section, is 

\l (cZc -j- c/j + d C| 6/x -f* 2 b ’ (1) 

Next, suppose that the diagonal runs from A to B 1 . In this case, 
through Z?! draw the planes B l E x E, B x A E , and B x E F (not rep¬ 
resented in the figure), dividing the half-section again into three 
quadrangular pyramids, having for their common vertex the point 
, and for their bases the planes AA i E l E ) E E 1 F 1 F, and A B FE. 
For the areas of these bases we have 

Area of AA 1 E 1 E = ^EE l x (A E + A x Ed) = \ l (c c t ), 

“ il EE 1 F 1 F = EF x EE t =£61, 

“ “ABFE + 

and for the perpendiculars from Bi on these bases, produced when 
accessary, 









102 


EARTH-WORK. 


Perpendicular = E x G 1 = d lf 

“ “ EE,F X F = B x G x =h x , 

“ « ABFE =EE t = /. 

TIjii (Tab. X. 52) the solidities of the three pyramids are 

B x - A A x EiE = X \ l[c + ci) —\l (<7 x c -{- c7 x c x ), 

Bi — EE x F x F — J Zt x X % b 7 == \lbh l , 

B x - AB FE = |7 X + + J6A). 

Their sum, or the solidity of the half-section, is 

6 1 (d c -f- t7 x c x -f- g/ x c -f- 6 7j x -j- ^ bh). (2) 

"We have thus found the solidity of the half-section for both direc 
tions of the diagonal. Let us now compare the results (1) and (2), 
and express them, if possible, by one formula. For this purpose let 
(1) be put under the form 

g l \dc + g/ x c x -f- d c 1 -f- ^ b (h + h x -f- A)], 
and (2) under the form 

11 [c?c + d l c l + di c -f- \ b (h -f- 7t x + 7i x )]. 

The only difference in these two expressions is, that d c x and the last 
h in the first, become d x c and 7i x in the second. But in the first case, 
Cj and h are the heights at the extremities of the diagonal, and d is the 
distance out corresponding to h ; and in the second case, c and h 1 are 
the heights at the extremities of the diagonal, and g7 x is the distance 
out corresponding to 7j x . Denote the centre height touched by the diagonal 
by C, the side height touched by the diagonal by H, and the distance out cor ■* 
responding to the side height II by D. We may then express both dc t 
and g7j c by D C ' and both h and hi by H\ so that the solidity of the 
half-section on the right of the centre line, whichever way the diago* 
nal runs, may be expressed by 

\l \dc + d x c x *f- D C 2 ^ {J 1 “f" + ZZ)"j. (3) 

To obtain the contents of the portion on the left of the centre line, 
we designate the quantities on the left by the same letters used for cor* 
responding quantities on the right, merely attaching a (') to them to 
distinguish them. Thus the side heights are h' and h\ , and the dis* 
tances out d' and d\ , while Z), ( 7 , and H become D’, C 1 , and IV. 
The solidity of the half-section on the left may therefore be taken di¬ 
rectly from (3), which will become 


CENTRE AND SIDE HEIGHTS GIVEN. 


103 


* l\d*c + d\ Cl + D< C' + ±b (h' + h\ + H<)]. (4) 

Finally, by uniting (3) and (4), we obtain the following formula for 
the solidity of the whole section between two stations 

S = \l{{d + d')c + {d l + d’ x )c 1 + DC+£»Ci + ±b{h + 

h + H+V + h't + H')]. 

Example. Given l = 100, b = 18, and the remaining data, as ar 
ranged in the first six columns of the following table. The first col¬ 
umn gives the stations; the fourth gives the centre heights, namely, 
c = 13.6 and c x = 8 ; the two columns on the left of the centre heights 
give the side heights and distances out on the left of the centre line of 
the road, and the two columns on the right of the centre heights give 
the side heights and distances out on the right. The direction of the 
diagonals is marked by the oblique lines drawn from h’ = 8 to c x = 8 
and from c = 13.6 to =' 12. 


Sta. 

cl'. 

h'. 

c. 

h. 

d. 

d + d'. 

(d + d')c . 

D ' C'. 

B C. 

0 

21 

8\ 

13.6 ''■s 

10 

24 

45 

612 



1 

15 

4 

> 8.0 

^12 

27 

42 

336 

168 

367.2 


12 12 168 
20 367.2 

54 X 9 = 486 

6)1969.20 

32820. 

To apply the formula, the distances out at each station are added 
together, and their sum placed in the seventh column; these sums, 
multiplied by the respective centre heights, are placed in the eighth 
column ; the product of d 1 = 21 (which is the distance out correspond¬ 
ing to the side height touched by the left-hand diagonal) by c t = 8 
I (which is the centre height touched by the same diagonal) is placed 
m the ninth column, and the similar product of d t = 27 by c = 13.6 
jis placed in the last column. The terms in the formula multiplied by 
| §• b are all the side heights, and in addition all the side heights touched 
by diagonals, or 8 + 4 + 10 + 12 + 8 + 12 = 54. Then by sub¬ 
stitution in the formula, we have S = g X 100 (612 + 336 + 168 -f 
367.2 + 9 X 54) = 32,820.cubic feet* 


* The example here given is the same as that calculated in Mr. Borden’s u Sys- 





























104 


EARTH-WORK. 


By applying the rule given in the note to § 121, we see that the sur¬ 
face on the left of the centre line in the preceding example is a plane; 
since 13.6 — 8:8 — 4 = 21 : 15. The diagonal on that side might, 
therefore, be taken either way, and the same solidity would be ob¬ 
tained. This may be easily seen by reversing the diagonal in this ex¬ 
ample, and calculating the solidity anew. The only parts of the for¬ 
mula affected by the change are D { C' and ^b H'. In the one case 
the sum of these terms is2lx 8 + 9X8, and in the other 15 X 13.6 
+ 9 X 4, both of which are equal to 240. 


123. IProH&SciBl. To find the solidity S of any number n of succes¬ 
sive sections of equal length. 

Solution. Let c, c x , c 2 , c 3 , &c. be the centre heights at the succes¬ 
sive stations; A, hi, h 2 , h 3 , &c. the right-hand side heights; h ', h\ , ldo , 
ld 3 , &c. the left-hand side heights; d, d t , d 2 ,d 3 , &c. the distances out 
on the right; and d', d\, d' 2 , d ' 3 , &c. the distances out on the left. 
Then the formula for the solidity of one section (§ 122) gives for the 
solidities of the successive sections 


S/[(cZ + d')c + (d t + d\) Cl -f D C + D' C' + ib {h + A x -f H+ 
h' + h\+H')], 

11 [(^i + d' i) c t -f- (do -f- d' 2 ) Co + D x C x -f- D\ C'i -j- 2 A (hi -f-A 2 -f- 

Hi + h'i + h’ 2 + H'i)\, 

11 \(d 2 + d'o) Co -f- (d 3 -f- d ' 3 ) c 3 -f- To C 2 -f- T'o C\ + ^ b (h 2 -f- A 3 -f- 

Ho -j- ldo -j- Id 3 + H' 2 Y \, 

and so on, for any number of sections. Tor the solidity of any num¬ 
ber n of sections, we should have n l multiplied by the sum of n paren¬ 
theses formed as those just given. Hence 


S=\l 


(d-\-d’)c-\- 2 d'i)ci+ 2 (d 2 + d' 2 ) c 2 ... + ( d n + d' n )c n 

-\-DC-\- D'C' -\-DyCifi- T'iC'i -\-T 2 Co-\- T' 2 C’ 2 -f & c. 




h -f- 2 h x -f- 2 h 2 .-j- h n + H -f- Hi + Ho -j- &.c. 

-f- A'-f- 2 A'x+ 2 h ’ 2 ... -f- Id n + H-\-H\-\-H' o -j- &c. 


tem of Useful Formulae, &c.,” page 187. It will be seen, that his calculation makes 
the solidity 32,460 cubic feet, which is 360 cubic feet less than the result above. 
This difference is owing to the omission, by Mr. Borden’s method, of a pyramid in¬ 
closed by the four pyramids, into which the upper portion of the right-hand half 
section is by that method divided. 





CENTRE AND SIDE HEIGHTS GIVEN. 


105 


Example. Given l — 100, b = 28, and the remaining data as given 
in the first six columns of the following table. 


Sta. 

d 1 . 

h'. 

c. 

0 

17 

K 

2 

1 

18.5 

3 


2 

20 

4 

^- 5 ~- 

3 

23 

6 


4 

21.5 

5"" 


5 

20 

4"^ 

_6 

6 

15.5 

1 

4 


25 


22 

22 

~69 

102 


A. 

d. 

d + d'. 

2 

17 

34 

5 

21.5 

40 

^6 

23 

43 

^8 

26 

49 

>7 

24.5 

46 

4 

20 

40 

3 

18.5 

34 


35 


30 

37 

102 


(d + d 1 ) c. 

D'C'. 

D C. 

68 



160 

68 

43 

215 

80 

92 

294 

115 

130 

276' 

129 

147 

240 

120 

147 

136 

93 

80 

1389 

605 

639 

1185 

605 


♦ 

639 

2394 

6)6212 


103533 cubic feet. 


171 X 14 = 2394 


The data in this table are arranged precisely as in the example for cal¬ 
culating one section (§ 122), and the remaining columns are calculated 
as there shown. Then, to obtain the first line of the formula, add all 
the numbers in the column headed (d-\- d 1 ) c, making 1389, and after¬ 
wards all the numbers except the first and the last, making 1185. 
The next line of the formula is the sum of the columns D' C' and 
D C, which give respectively 605 and 639. To obtain the first line of 
the quantities multiplied by ^b, add all the numbers in column /t, 
making 35, next all the numbers except the first and the last, making 
30, and lastly all the numbers touched by diagonals (doubling any one 
touched by two diagonals), making 37. The second line of the quan¬ 
tities multiplied by % b is obtained in the same way from the column 
marked h f . The sum of these numbers is 171, and this multiplied by 
16 = 14 gives 2394. We have now for the first line of the formula 
1389 -f- 1185, for the second 605 -f 639, and for the remainder 2394. 

ioo 

By adding, these together, and multiplying the sum by §1 = -g- , we 
get the contents of the six sections in feet. 

124. When the section is partly in excavation and partly in embank¬ 
ment, the preceding formulae are still applicable ; but as this applica¬ 
tion introduces minus quantities into the calculation, the following 
method, similar in principle, is preferable. 

125. Problem* Given the widths of an excavation at the road-bed 

6 




























106 


EARTH-WORK. 


A F — w and A 1 F x = w l {Jig- 53), the side heights h and h l , the length 
of the section l , and the direction of the diagonal , to find the solidity S of 
the excavation , when the section is partly in excavation and partly in em¬ 
bankment. 



Solution. Suppose, first, that the surface is divided into two trian 
gles by the diagonal B A v Through B draw the plane BA 1 F l , 
dividing that part of the section which is in excavation into two pyra¬ 
mids B- A Ai F l F and B - A l B 1 F 1 , the solidities of which are 

B - A A x F r F = \h X ^l{w w x ) =1 l{wh A- u^h), 
B-A i B l F l =hlX%w 1 h 1 = llw 1 h l . 

The whole solidity is, therefore, 

S — \ l {wh w l h l w l h). 

Next, suppose the dividing diagonal to run from A to B x . Thi’ough 
B x draw a plane B x A F (not represented in the figure), dividing the 
excavation again into two pyramids, of which the solidities are 

B x — A Ai F x F — J hi X 2 l {w -f- w x ) = \ l (w h t w t A x ), 
Bi-ABF =hlX h wh = \lwh. 

The whole solidity is, therefore, 

S = § l {wh + w x hi -f- w h^. 

The only difference in these two expressions is, that h in the first 
becomes w hi in the second. But in the first case the diagonal touch¬ 
es Wi and h, and in the second case it touches w and h v If, then, we 
designate the width touched by the diagonal by W, and # the height 
touched by the diagonal by H , we may express both w x h and w h x by 
WIT) so that the solidity in either case may be expressed by 


CENTRE AND SIDE HEIGHTS GIVEN. 


107 


^ S = ll [wh + w x h x + WE). 

Corollary. When several sections of equal length succeed one 
another, the whole may be calculated together. For this purpose, the 
preceding formula gives for the solidities of the successive 1 sections 

ll(wh +w 1 A 1 +W\ff), 

5 1 [wi hi + K + Wi Hi), 

i l [w 2 h 2 -j- -J- Wo Ho), 

and so on for any number of sections. Hence for the solidity of any 
number n of sections we should have 

S = g l (w h -j- 2 uq hi -f- 2 w 2 ho .... -j- io,i hn -J- WH -{- TFj. H\ *4- 
W 2 Ho + &c.) 

Example. Given l = 100, and the remaining data as given in the 
first three columns of the following table. 


Station. 

XV. 

h. 

iv h. 

WH. 

0 

2 

/I 

2 


1 

8< 

6 

48 

8 

2 

10^ 

^7 

70 

56 

3 

13^ 

7 

91 

70 

4 

9 

""" 4 

36 

52 


247 186 


209 

186 

6)642 

10700. 

The fourth column contains the products of the several widths by 
the corresponding heights, and the next column the products of those 
widths and heights touched by diagonals. The sum of the products 
in the fourth column is 247, the sum of all but the first and the last is 
209, and the sum of the products in the fifth column is 186. These 
three sums are added together, multiplied by 100, and divided by 6, 
according to the formula. This gives the solidity of the four sections 
= 10700 cubic feet. 

126. When the excavation does not begin on a line at right angles 
to the centre line, intermediate stations are taken where the excava* 
tion begins on each side of the road-bed, and the section may be calcu- 














EARTH-WORK. 


10S 

Iated as a pyramid, having its vertex at the first of these points, and 
for its base the cross-section at the second. The preceding method 
gives the same result, since w and h in this case become 0, and reduce 
the formula to S =■= 1 l w l h x . The same remarks apply to the end of 
an excavation. 


C. Ground very Irregular . 

127. Problem. To find the solidity of a section, when the ground 
is very irregular. 



Solution. Let A HB FE - A x CD B x F 1 E i (fig. 54) represent one 
side of a section, the surface of which is too irregular to be divided 
into two planes. Suppose, for instance, that the ground changes at 
H , C , and D , making it necessary to divide the surface into five trian¬ 
gles running from station to station.* Let heights be taken at II , C, 
and D , and let the distances out of these points be measured. If now 
we suppose the earth to be excavated vertically downward through 
the side line BB t to the plane of the road-bed, we may form as many 
vertical triangular prisms as there are triangles on the surface. This 
will be made evident by drawing vertical planes through the sides 


* It will often be necessary to introduce intermediate stations, in order to maks 
♦he subdivision into triangles more conveniently and accurately. 






GROUND VERY IRREGULAR. 


109 


A C , II C, HD. and H B x . Then the solidity of the half-section will he 
equal to the sum of these prisms , minus the triangular mass B F G- 

B x F x G l . 

The horizontal section of the prisms may be found from the distan¬ 
ces out and the length of the section, and the vertical edges or heights 
are all known. Hence the solidities of these prisms may be calculated 
by § 115. 

To find the solidity of the portion B F G-B x F x Gi , which is to 
be deducted, represent the slope of the sides by s (§ 102), the heights 
at B and B x by h and h x , and the length of the section by l. Then 
we have F G = s h, and F x G x = s h x . Moreover, the area of B F G 
— f s /r, and that of B x F x G x — £sh x 2 . Now as the triangles B F G 
and B x F x G x are similar, the mass required is the frustum of a pyra¬ 
mid, and the mean area is s h 2 x j s h x 2 = % s h h x . Then 
(Tab. X. 53) the solidity is BF G-B x F x G x =}ls {h 2 +h x 2 -\-hh x ). 

Example. Given l — 50, h ==18, s = §, the heights at A, H, and B 
respectively 4, 7, and 6, the distances A H = 9 and IIB = 9, the 
heights at A x , C , Z), and B x respectively 6, 7, 9, and 8, and the distan¬ 
ces A x C = 4, CD — 5, and D B v = 12. Then the horizontal sec¬ 
tion of the first prism adjoining the centre line is ^l X A x C, since the 
distance A x C is measured horizontally; and the mean of the three 
heights is £ (4 + 6 + 7) = £ X 17. The solidity of this prism is 
therefore X A x C X £ X 17 = 5 / X 4 X 17, that is, equal to \ l 
multiplied by the base of the triangle and by the sum of the heights. 
In this way we should find for the solidity of the five prisms 

U(4 X 17 + 9 X 18 + 5 X 23+ 12 X 24 + 9 X 21) = \l X 822. 

For the frustum to be deducted, we have 

. ll X i(6 2 + 8 2 + 6 x 8) = u X 222 . 

Hence the solidity of the half-section is 

ll (822 — 222) = | X 50 X 600 = 5000 cubic feet. 

128. Let us now examine the usual method of calculating excava¬ 
tion, when the cross-section of the ground is not level. This method 
consists, first, in finding the area of a cross-section at each end of the 
mass ; secondly, in finding the height of a section, level at the top , 
equivalent in area to each of these end sections; thirdly, in finding 
from the average of these two heights the middle area of the mass; 



MO 


EARTH-WORK. 


and, lastly, in applying the prismoidal formula to find the contents 
The heights of the equivalent sections level at the top may be found 
approximately by Trautwine’s Diagrams,* or exactly by the following 
method. Let A represent the area of an irregular cross-section, b the 
width of the road-bed, and s the slope of the sides. Let x be the re¬ 
quired height of an equivalent section level at the top. The bottom 
of the equivalent section will be b, the top 6 + 2 s x, and the area will 
be the sum of the top and bottom lines multiplied by half the height or 
£ x (2 b + 2sx) = sx 2 -f- bx. But this area is to be equal to A 
Therefore, s x 2 b x = A, and from this equation the value of x may 
be found in any given case. 

According to this method, the contents of the section already calcu¬ 
lated in § 122 will be found thus. Calculating the end areas, w r e find 
the first end area to he 387 and the second to be 240. Then as s is 
here \ and b = 18, the equations for finding the heights of the equiva¬ 
lent end sections will be §:r 2 -f- 18a: = 387, and %x 2 18 :r = 240. 

Solving these equations, we have for the height at the first station 
x — 11.146, and at the second, x = 8. The middle area will, there¬ 
fore, have the height ^ (11.146 -f- 8) = 9.573, and from this height the 
middle area is found to be 309.78. Then by the prismoidal formula 
(S 113) the solidity will be S = \ X 100 (387 -f 240 -{- 4 X 309.78) 
-- 31102 cubic feet. 

But the true solidity of this section was found to he 32820 cubic 
feet, a difference of 1718 feet. The error, of course, is not in the pris¬ 
moidal formula, but in assuming that, if the earth were levelled at the 
ends to the height of the equivalent end sections, the intervening earth 
might be so disposed as to form a plane between these level ends, thus 
reducing the mass to a prismoid. This supposition, however, may 
sometimes be very far from correct, as has just been shown. If the 
diagonal on the right-hand side in this example were reversed, that is, 
if the dividing line w r ere formed by a depression, the true solidity 
found by § 122 w r ould be 29600 feet; whereas the method by equiva¬ 
lent sections would give the same contents as before, or 1502 feet too 
much. 

D. Correction in Excavation on Curves 

129. In excavations on curves the ends of a section are not parallel „ 


* A New Method of Calculating the Cubic Contents of Excavations and Embank 
ments by the aid of Diagrams. By John C, Trautwine. 




CORRECTION IN EXCAVATION ON CURVES. 


Ill 


to each other, but converge towards the centre of the curve. A section 
between two stations 100 feet apart on the centre line will, therefore, 
measure less than 100 feet on the side nearest to the centre of the 
curve, and more than 100 feet on the side farthest from that centre. 
Now in calculating the contents of an excavation, it is assumed that 
the ends of a section are parallel, both being perpendicular to the chord 
of the curve. Thus, let figure 55 represent the plan of two sections oi 



an excavation, E F G being the centre line, A L and CM the extreme 
side lines, and O the centre of the curve. Then the calculation of the 
first section would include all between the lines A j C x and B x Di’ y 
<vhile the true section lies between A C and B D. In like manner, the 
calculation of the second section would include all between HK and 
NP, while the true section lies between BD and L M. It is evident, 
therefore, that at each station on the curve, as at F y the calculation is 
too great by the wedge-shaped mass represented by KFD X , and too 



•mall by the mass represented by B X FU These masses balance 










112 


EARTH-WORK. 


each other, when the distances out on each side of the centre line are 
equal, that is, when the cross-section may be represented by A DFRE 
(fig. 56). But if the excavation is on the side of a hill, so that the 
distances out differ very much, and the cross-section is of the shape 
A D FB E , the difference of the wedge-shaped masses may require 
consideration. 

130. Problem. Given the centre height c, the greatest side height h, 
the least side height hthe greatest distance out d, the least distance out d', 
and the width of the road-bed b, to find the correction in excavation C, at 
any station on a curve of radius R or deflection angle D. 

Solution. The correction, from what has been said above, is a trian¬ 
gular prism of which B FR (fig. 56) is a cross-section. The height of 
this prism at B (fig. 55) is B x H ) the height at R is R X S, and the height 
at F is 0. B x H and R l S , being very short, are here considered 
straight lines. Now we have the cross-section B FR = FB EG — 
FREG = [$cd -+ ibh) — {±cd< + ibh’) = }c(d — d<) + 
i b (h — h’). To find the height B x H, we have the angle B FH — 
BFBi = D, and therefore B x H — 2 IIF sin. D = 2d sin. D. In 
like manner, R A S = KD X = 2 KF sin. Z> = 2tHsin. D. Then 
since the height at F is 0, one third of the sum of the heights of the 
prism will be g (d +- c?')sin. Z>, and the correction, or the solidity of 
the prism, will be (§ 115) 

G?" C= [i c (d—d') + £b(h — A')] X | (d + d>) sin. D. 

When R is given, and not Z), substitute for sin. D its value (§ 9 ) 
50 

sin. D — x . The correction then becomes 

Es*" C = a c (d - d ') + i b (h — t'H x 100 . 

3 R 

This correction is to be added , when the highest ground is on the 
convex side of the curve, and subtracted , when the highest ground is on 
the concave side. At a tangent point, it is evident, from figure 55, that 
the correction will be just half of that given above. 

Example. Given c = 28, h — 40, h' = 16, d — 74, d r = 38, b = 28, 
and R = 1400, to find C. Here the area of the cross-section BFR 
28 28 

2 " (74 — 38) + (40 — 16) = 672, and one third of the sum of the 

, . , _ , . . 100(74 + 38) 8 ^ „ 8 

heights of the prism is ~ 3 ~ x l 400— ~ 3 • Hence C = 672 X 3 *«=* 

(792 cubic feet. 




CORRECTION IN EXCAVATION ON CURVES. 


113 


131. When the section is partly in excavation and partly in em¬ 
bankment, the cross-section of the excavation is a triangle lying 
wholly on one side of the centre line, or partly on one side and partly 
on the other. The surface of the ground, instead of extending from 
B to D (fig. 56), will extend from B to a point between G and E , or 
to a point between A and G. In the first case, the correction will be 
a triangular prism lying between the lines B l F and ELF (fig. 55), but 
not extending below the point F. In the second case, the excavation 
extends below F , and the correction, as in § 129, is the difference be¬ 
tween the masses above and below F. This difference may be ob¬ 
tained in a very simple manner, by regarding the mass on both sides 
of F as one triangular prism the bases of which intersect on the line 
GF (fig. 56), in which case the height of the prism at the edge be¬ 
low F must be considered to be minus , since the direction of this edge, 
referred to either of the bases, is contrary to that of the two others. 
The solidity of this prism will then be the difference required. 

132. Given the width of the excavation at the road-bed 

w, the width of the road-bed b, the distance out d, and the side height h , to 
find the correction in excavation C, at any station on a curve of radius R 
or defection angle D, when the section is partly in excavation and partly in 
embankment. 

Solution. When the excavation lies wholly on one side of the centre 
line, the correction is a triangular prism having for its cross-section 
the cross-section of the excavation. Its area is, therefore, ^ w h. The 
height of this prism at B (fig. 56) is (§ 130) B ± II = 2 HF sin. D — 
2 d sin. D. In a similar manner, the height at E will be 2 G E sin. D 

— b sin. D, and at the point intermediate between G and E , the dis¬ 
tance of which from the centre line is % b — w, the height will be 
2 (£b — w) sin .D— ( b — 2 w) sin. D. Hence, the correction, or the solid¬ 
ity of the prism, will be (§115) C = ^whx\ [2d-\-b-\-b — 2 w) sin. D 

— ^ w li X | [d + b — tv) sin. D. 

When the excavation lies on both sides of the centre line, the cor¬ 
rection, from what has been said above, is a triangular prism having 
also for its cross-section the cross-section of the excavation. Its area 
will, therefore, be ^ tv h. The height of this prism at B is also 2 d sin. IJ , 
and the height at E, b sin. D; but at the point intermediate between A 
and G , the distance of which from the centre line is w — \b, the height 
will be 2 (tv — &b) sin. D — (2 w — b) sin. D. As this height is to 
be considered minus, it must be subtracted from the others, and the 
correction required will be C = jto h X J (2 d -f- b — 2 to -f b) sin. D 


114 


EAETII-WORK. 


c=iw/iX | (d -J- 6 — iv) sin. D. Hence, in all cases, when the sec¬ 
tion is partly in excavation and partly in embankment, we have the 
formula 

ISP C — £ iv h X | (d -f- b — w) sin. D. 

When R is given, and not Z), substitute for sin. D its value (§ 9) 
50 

sin. D — jv . The correction then becomes 
r-sp n i t .. 100 [d-\~b — w) 

I5P C — h w h X - v ■ ^ -- . 

3 R 

This correction is to be added , when the highest ground is on the 
convex side of the curve, and subtracted when the highest ground is on 
the concave side. At a tangent point the correction will be just half 
of that given above. 


Example. Given w = 17, b — 30, d — 51, h = 24, and R — 1600, 
to find C. Here the area of the cross-section is \wh — 17 X *2 = 


100 (d + b—ic) 


204, and one third of the sum of the heights of the prism is- 

4 4 

= g. Hence C = 204 X j = 272 cubic feet. 


100 (51 + 30 —17) 
3 X 1600 


133. The preceding corrections (§ 130 and § 132) suppose the length 
of the sections to be 100 feet. If the sections are shorter, the angle 
B FII (fig. 55) may be regarded as the same part of D that F G is ol 
100 feet, and FB as the same part of D that E F is of 100 feet. 
The true correction may then be taken as the same part of C that the 
sum of the lengths of the two adjoining sections is of 200 feet. 





TABLE I. 


RADII, ORDINATES, DEFLECTIONS, 


AND 


ORDINATES FOR CURVING RAILS. 

« 

Formula for Radii, $ 10; for Ordinates, § 25; for Deflections, § l‘J 
for Curving Rails, § 29. 


116 TABLE I. RADII, ORDINATES, DEFLECTIONS, 


Degree. 

Radii. 

Ordinates. 

Tangent 

Deflec¬ 

tion. 

Chord 

Deflec¬ 

tion. 

Ordinates for 
Rails. 

12*. 

25. 

CO 

60. 

CO 

20. 

o / 

0 5 

68754.94 

.008 

.014 

.017 

.018 

.073 

.145 

.001 

.001 

10 

34377.48 

.016 

.027 

.034 

.036 

.145 

.291 

.001 

.001 

15 

22918 33 

.024 

.041 

.051 

.055 

.218 

.436 

.002 

.002 

20 

17188.76 

.032 

.055 

.068 

.073 

.291 

.582 

.002 

.003 

25 

13751.02 

.040 

.068 

.085 

.091 

.364 

.727 

.003 

.004 

30 

11459.19 

.048 

.082 

.102 

.109 

.436 

.873 

.004 

.004 

35 

9322.18 

.056 

.095 

.119 

.127 

.509 

1.018 

.004 

.005 

40 

8594.41 

.064 

.109 

.136 

.145 

.582 

1.164 

.005 

.006 

45 

7639.49 

.072 

.123 

.153 

.164 

.654 

1.309 

.005 

.007 

50 

6875.55 

.080 

.136 

.170 

.182 

.727 

1.454 

.006 

.007 

55 

6250.51 

.087 

.150 

.187 

.200 

.800 

1.600 

.006 

.008 

1 0 

5729.65 

.095 

.164 

.205 

.218 

.873 

1.745 

.007 

.009 

5 

5238.92 

103 

.177 

.222 

.236 

.945 

1.891 

.008 

.009 

10 

4911.15 

.111 

.191 

.239 

.255 

1.018 

2.036 

.008 

.010 

15 

4583.75 

.119 

.205 

.256 

.273 

1.091 

2.182 

.009 

.011 

20 

4297.28 

.127 

.218 

.273 

.291 

1.164 

2.327 

.009 

.012 

25 

4044.51 

.135 

.232 

.290 

.309 

1.236 

2.472 

.010 

.012 

30 

3819.83 

.143 

.245 

.307 

.327 

1.309 

2.618 

.011 

.013 

35 

3618.80 

.151 

.259 

.324 

.345 

1 382 

2.763 

.011 

.014 

40 

3437.87 

.159 

.273 

.341 

.364 

1.454 

2.909 

.012 

.015 

45 

3274.17 

.167 

.286 

.358 

.332 

1.527 

3.054 

.012 

.015 

50 

3125.36 

.175 

.300 

.375 

.400 

1.600 

3.200 

.013 

.016 

55 

2989.48 

.183 

.314 

.392 

.418 

1.673 

3.345 

.014 

017 

3 0 

2864.93 

.191 

.327 

.409 

.436 

1.745 

3.490 

.014 

.017 

5 

2750.35 

.199 

.341 

.426 

.455 

1.818 

3.636 

.015 

.018 

10 

2644.58 

.207 

.355 

.443 

.473 

1.891 

3.781 

.015 

.019 

15 

2546.64 

.215 

.368 

.460 

.491 

1.963 

3.927 

.016 

.020 

20 

2455.70 

.223 

.382 

.477 

.509 

2.036 

4.072 

.016 

.020 

25 

2371.04 

.231 

.395 

.494 

.527 

2.109 

4.218 

.017 

.021 

30 

2292.01 

.239 

.409 

.511 

.545 

2.181 

4.363 

.018 

.022 

35 

2218.09 

.247 

.423 

.528 

.564 

2.254 

4.508 

.018 

.023 

40 

2148.79 

.255 

.436 

.545 

.582 

2.327 

4.654 

.019 

.023 

45 

2083.68 

.263 

.450 

.562 

.600 

2.400 

4.799 

.019 

.024 

50 

2022.41 

.270 

.464 

.580 

.618 

2.472 

4.945 

.020 

.025 

55 

1964.64 

.278 

.477 

.597 

.636 

2.545 

5.090 

.021 

.025 

3 0 

1910.08 

.28 6 

.491 

.614 

.655 

2.618 

5.235 

.021 

.026 

5 

1858.47 

•294 

.505 

.631 

.673 

2.690 

5.381 

.022 

.027 

10 

1809.57 

.302 

.518 

.648 

.691 

2.763 

5.526 

.022 

.028 

15 

1763 18 

.310 

.532 

.665 

.709 

2.836 

5.672 

.023 

.028 

20 

1719.12 

.318 

.545 

.682 

.727 

2.908 

5.817 

.024 

.029 

25 

1677.20 

.326 

.559 

.699 

.745 

2.981 

5.962 

.024 

.030 

30 

1637.28 

.334 

.573 

.716 

.764 

3.054 

6.108 

.025 

.031 

35 

1599.21 

.342 

.586 

.733 

.782 

3.127 

6.253 

.025 

.031 

40 

1562.88 

.350 

.600 

.750 

.800 

3.199 

6.398 

.026 

.032 

45 

1528.16 

.358 

.614 

.767 

.818 

3.272 

6.544 

.027 

.033 

50 

1494.95 

.366 

.627 

.784 

.836 

3.345 

6.689 

.027 

.033 

55 

1463.16 

.374 

.641 

.801 

.855 

3.417 

6.835 

.028 

.034 

4 0 

1432.69 

.382 

.655 

.818 

.873 

3.490 

6.980 

.028 

.035 

5 

1403.46 

.390 

.668 

.835 

.891 

3.563 

7.125 

.029 

.036 

10 

1375.40 

.398 

.682 

.852 

.909 

3.635 

7.271 

.029 

.036 

15 

1348.45 

.406 

.695 

.869 

.927 

3.708 

7.416 

.030 

.037 

20 

1322.53 

.414 

.709 

.886 

.945 

3.781 

7.561 

.031 

.03S 

25 

1297.58 

.422 

.723 

.903 

.964 

3.853 

7.707 

.031 

.039 

30 

1273.57 

.430 

.736 

.921 

.982 

3.926 

7.852 

.032 

.039 

35 

1250.42 

.438 

.750 

.938 

1.000 

3.999 

7.997 

.032 

.040 

40 

1228.11 

.446 

.764 

.955 

1.018 

4.071 

8.143 

.033 

.041 

45 

1206.57 

.454 

.777 

.972 

1.036 

4.144 

8.288 

.034 

.011 

50 

1185.78 

.462 

.791 

.989 

1.055 

4.217 

8.433 

.034 

.042 

55 

1165.70 

.469 

.805 

1.006 

1.073 

4.289 

8.579 

.035 

.043 

5 0 

1146.28 

.477 

.818 

1.023 

1.091 

4.362 

8.724 

.035 

.044 



























AND ORDINATES FOR CURVING RAILS 


117 


Degree. 

Radii. 

Ordinates. 

Tangent 

Deflec¬ 

tion. 

Chord 

Deflec¬ 

tion. 

Ordinates for 
Rails. 

12*. 

25. 

CO ' 

50. 

18. 

20. 

O 1 

5 5 

1127.50, 

.485 

.832 

1.040 

1.109 

4.435 

8.869 

.036 

.044 

10 

1109.33 

.493 

.846 

1.057 

1.127 

4.507 

9.014 

.037 

.045 

15 

1091.73 

.501 

.859 

1.074 

1.146 

4.580 

9.160 

.037 

.046 

20 

1074.68 

.509 

.873 

1.091 

1.164 

4.653 

9.305 

.03S 

.047 

25 

1U58.16 

.517 

.887 

1.108 

1.182 

4.725 

9.450 

.038 

.047 

30 

1042.14 

.525 

.900 

1.125 

1 200 

4.798 

9.596 

.039 

.048 

35 

1026.60 

.533 

.914 

1.142 

1.218 

4.870 

9.741 

.039 

.049 

40 

1011.51 

.541 

.928 

1.159 

1.237 

4.943 

9.886 

.040 

.049 

45 

996.87 

.549 

.941 

1.176 

1.255 

5.016 

10.031 

.041 

.050 

50 

982.64 

.557 

.955 

1.193 

1.273 

5.088 

10.177 

.041 

.051 

55 

968.81 

.565 

.968 

1.210 

1.291 

5.161 

10.322 

.042 

.052 

6 0 

955.37 

.573 

.982 

1.228 

1.309 

5.234 

10.467 

.042 

.052 

5 

942.29 

.581 

.996 

1.245 

1.327 

5.306 

10.612 

.043 

.053 

10 

929.57 

.589 

1.009 

1.262 

1.346 

5.379 

10.758 

.044 

.054 

15 

917.19 

.597 

1.023 

1.279 

1.364 

5.451 

10.903 

.044 

.055 

20 

905.13 

.605 

1.037 

1.296 

1.382 

5.524 

11.048 

.045 

.055 

25 

893.39 

.613 

1.050 

1.313 

1.400 

5.597 

11.193 

.045 

.056 

30 

881.95 

.621 

1.061 

1.330 

1.418 

5.669 

11.339 

.046 

.057 

35 

870.79 

.629 

1.078 

1.347 

1.437 

5.742 

11.484 

.047 

.057 

40 

859.92 

.637 

1.091 

1.364 

1.455 

5.814 

11.629 

.047 

.058 

45 

849.32 

.645 

1.105 

1.381 

1.473 

5.887 

11.774 

.048 

.059 

50 

838.97 

.653 

1.118 

1.398 

1.491 

5.960 

11.919 

.048 

.060 

55 

828.88 

.661 

1.132 

1.415 

1.510 

6.032 

12.065 

.049 

.060 

7 0 

819.02 

.669 

1.146 

1.432 

1.528 

6.105 

12.210 

.049 

061 

5 

809.40 

.677 

1.159 

1.449 

1.546 

6.177 

12.355 

.050 

.062 

10 

800.00 

.685 

1.173 

1.466 

1.564 

6.250 

12.500 

.051 

.063 

15 

790.81 

.693 

1.187 

1.483 

1.582 

6.323 

12.645 

.051 

.063 

20 

781.84 

.701 

1.200 

1.501 

1.600 

6.395 

12.790 

.052 

.064 

25 

773.07 

.709 

1.214 

1.517 

1.619 

6.468 

12.936 

.052 

.065 

30 

764.49 

.717 

1.22S 

1.535 

1.637 

6.540 

13.081 

.053 

.065 

35 

756.10 

.725 

1.242 

1.552 

1.655 

6.613 

13.226 

.054 

.066 

40 

747.89 

.733 

1.255 

1.569 

1.67.3 

6.685 

13.371 

.054 

.067 

45 

739.86 

.740 

1.269 

1.586 

1.691 

6.758 

13.516 

.055 

.068 

50 

732.01 

.748 

1.283 

1.603 

1.710 

6.831 

13.661 

.055 

.068 

55 

724.31 

.756 

1.296 

1.620 

1.728 

6.903 

13.806 

.056 

.069 

8 0 

716.78 

.764 

1.310 

1.637 

1.746 

6.976 

13.951 

.057 

070 

5 

709.40 

.772 

1.324 

1.654 

1.764 

7.048 

14.096 

.057 

.070 

10 

702.18 

.780 

1.337 

1.671 

1.7S2 

7.121 

14.241 

.058 

.071 

15 

695.09 

.788 

1.351 

1.688 

1.801 

7.193 

14.387 

.058 

.072 

20 

688.16 

.796 

1.365 

1.705 

1.819 

7.266 

14.532 

.059 

.073 

25 

681.35 

.804 

1.378 

1.722 

1.837 

7.338 

14.677 

.059 

.073 

30 

674.69 

.812 

1.392 

1.739 

1.855 

7.411 

14.822 

.060 

.074 

35 

668.15 

.820 

1.406 

1.757 

1.873 

7.483 

14.967 

.061 

.075 

40 

661.74 

.828 

1.419 

1.774 

1.892 

7.556 

15.112 

.061 

.076 

45 

655.45 

.836 

1.433 

1.791 

1.910 

7.628 

15.257 

.062 

.076 

50 

649.27 

.844 

1.447 

1.808 

1.928 

7.701 

15.402 

.062 

.077 

55 

643.22 

.852 

1.460 

1.825 

1.940 

7.773 

15.547 

.063 

.078 

9 0 

637.27 

.860 

1.474 

1.842 

1.965 

7.846 

15.692 

.064 

.078 

5 

631.44 

.868 

1.488 

1.859 

1.9S3 

7 918 

15.837 

.064 

.079 

10 

625.71 

.876 

1.501 

1.876 

2.001 

7.991 

15.982 

.065 

.080 

15 

620.09 

.884 

1.515 

1.893 

2.019 

8.063 

16.127 

.065 

.081 

20 

614.56 

.892 

1.529 

1.910 

2.037 

8.136 

16.272 

.066 

.081 

25 

609.14 

.900 

1.542 

1.927 

2.056 

8.208 

16.417 

.0661 

.082 

30 

603.80 

.908 

1.556 

1.944 

2.074 

8.281 

16.562 

.067 

.083 

35 

598.57 

.916 

1.570 

1.961 

2.092 

8.353 

16.707 

.068 

.084 

40 

593.42 

.924 

1.583 

1.979 

2.110 

8.426 

16.852 

.068 

.084 

45 

588.36 

.932 

1.597 

1.996 

2.128 

8.498 

16.996 

.069 

.085 

50 

583.38 

.940 

1.611 

2.013 

2.147 

8.571 

17.141 

.069 

.086 

55 

578.49 

.948 

1.624 

2.030 

2.165 

8.643 

17.286 

.070 

.086 

10 0 

573.69 

.956 

1.638 

2.047 

2.183 

8.716 

17.431 

.071 

.087 






































118 TABLE I. RADII, ORDINATES, DEFLECTIONS, AC 


Degree. 

Radii. 

Ordinates. 

Tangent 

Deflec- 

tion. 

Chord 

Deflec¬ 

tion. 

Ordinates for 
Rails. 

12*. 

25. 

CO 

50. 

18. 

20. 

o 

10 

i 

10 

561.31 

.972 

1.665 

2.081 

2.219 

8.860 

17.721 

.072 

.089 


20 

555.23 

.988 

1.693 

2.115 

2.256 

9.005 

18.011 

.073 

.090 


30 

546.44 

1.004 

1.720 

2.149 

2.292 

9.150 

18.300 

.074 

.092 


40 

537.92 

1.020 

1.743 

2.184 

2.329 

9.295 

18.590 

.075 

.093 


50 

529.67 

1.036 

1.775 

2.218 

2.365 

9.440 

18.880 

.076 

.094 

11 

0 

521.67 

1.052 

1.802 

2.252 

2.402 

9.585 

19.169 

.078 

.096 


10 

513.91 

1.068 

1.830 

2.236 

2.438 

9.729 

19.459 

.079 

.097 


20 

506.38 

1.034 

1.857 

2.320 

2.475 

9.874 

19.748 

.080 

.099 


30 

499.06 

1.100 

1.884 

2.334 

2.511 

10.019 

20.038 

.081 

.100 


40 

491.96 

1.116 

1.912 

2.389 

2.547 

10.164 

20.327 

.032 

.102 


50 

485.05 

1.132 

1.938 

2.423 

2.584 

10.303 

20.616 

.084 

.103 

12 

0 

478.34 

1.148 

1.967 

2.457 

2.620 

10.453 

20.906 

.035 

.105 


10 

471.81 

1.164 

1.994 

2.491 

2.657 

10.597 

21.195 

.086 

.106 


20 

46o.46 

1.180 

2.021 

2.525 

2.693 

10.742 

21.484 

.037 

.107 


30 

459.28 

1.196 

2.019 

2.560 

2.730 

10.887 

21.773 

.088 

.109 


40 

453.26 

1.212 

2.076 

2.594 

2.766 

11.031 

22.063 

.089 

.110 


50 

447.40 

1.228 

2.104 

2.628 

2.803 

11.176 

22.352 

.091 

.112 

13 

0 

441.68 

1.244 

2.131 

2.662 

2.839 

11.320 

22.641 

.092 

.113 


10 

436.12 

1.260 

2.159 

2.697 

2.876 

11.465 

22.930 

.093 

.115 


20 

430.69 

1.277 

2.1S6 

2.731 

2.912 

11.609 

23.219 

.094 

.116 


30 

425.40 

1.293 

2.213 

2.765 

2.949 

11.754 

23.507 

.095 

.118 


40 

420.23 

1.309 

2.241 

2.799 

2.985 

11.898 

23.796 

.096 

.119 


50 

415.19 

1.325 

2.268 

2.833 

3.022 

12.043 

24.085 

.098 

.120 

14 

0 

410.28 

1.341 

2.296 

2.863 

3.058 

12.187 

24.374 

.099 

.122 


10 

405.47 

1.357 

2.323 

2.902 

3.095 

12.331 

24.663 

.100 

.123 


20 

400.78 

1.373 

2.351 

2.936 

3.131 

12.476 

24.951 

.101 

.125 


30 

396.20 

1.389 

2.378 

2.970 

3.168 

12.620 

25.240 

.102 

.126 


40 

391.72 

1.405 

2.406 

3.005 

3.204 

12.761 

25.528 

.103 

.123 


50 

387.34 

1.421 

2.433 

3.039 

3.241 

12.90S 

25.817 

.105 

.129 

15 

0 

333.06 

1.437 

2.461 

3.073 

3.277 

13.053 

26.105 

.106 

.131 


10 

378.88 

1.453 

2.438 

3.107 

3.314 

13.197 

26.394 

.107 

.132 


20 

374.79 

1.469 

2.515 

3.142 

3.350 

13.341 

26.682 

.103 

.133 


30 

370.78 

1.486 

2.543 

3.176 

3.387 

13.485 

26.970 

.109 

.135 


40 

366.86 

1.502 

2.570 

3.210 

3.423 

13.629 

27.258 

.110 

.136 


50 

363.02 

1.518 

2.598 

3.245 

3.460 

13.773 

27.547 

.112 

.133 

16 

0 

359.26 

1.534 

2.625 

3.279 

3.496 

13.917 

27.835 

.113 

.139 


10 

355.59 

1.550 

2.653 

3.313 

3.533 

14.061 

28.123 

.114 

.141 


20 

351.98 

1.566 

2.630 

3.347 

3.569 

14.205 

23.411 

.115 

.142 


30 

348.45 

1.582 

2.708 

3.382 

3.606 

14.349 

23.699 

.116 

.143 


40 

344.99 

1.598 

2.736 

3.416 

3.643 

14.493 

23.986 

.117 

.145 


50 

341.60 

1.615 

2.763 

3.450 

3.679 

14.637 

29.274 

.119 

.146 

17 

0 

333.27 

1.631 

2.791 

3.485 

3.716 

14.781 

29.562 

.120 

.148 


10 

335.01 

1.647 

2.818 

3.519 

3.752 

14.925 

29.850 

.121 

.149 


20 

331.82 

1.663 

2.846 

3.553 

3;789 

15.069 

30.137 

.122 

.151 


30 

328.63 

1.679 

2.873 

3.588 

3.825 

15.212 

30 425 

.123 

.152 


40 

325.60 

1.695 

2.901 

3.622 

3 862 

15.356 

30.712 

124 

.154 


50 

322.59 

1.711 

2.923 

3.656 

3.898 

15.500 

31.000 

.126 

.155 

18 

0 

319.62 

1.723 

2.956 

3.691 

3.935 

15.643 

31.287 

.127 

.156 


10 

316.71 

1.744 

2.983 

3.725 

3.972 

15.787 

31.574 

.123 

.158 


20 

313.86 

1.760 

3.011 

3.759 

4.003 

15.931 

31.861 

.129 

.159 


30 

311.06 

1.776 

3.039 

3.794 

4.045 

16.074 

32.149 

.130 

.161 


40 

303.30 

1.792 

3.066 

3.828 

4.081 

16.218 

32.436 

.131 

.162 


50 

305.60 

1.809 

3.094 

3.862 

4.118 

16.361 

32.723 

.133 

.164 

19 

0 

302.94 

1.825 

3.121 

3.897 

4.155 

16.505 

33.010 

.134 

.165 


10 

300.33 

1.841 

3.149 

3.931 

4.191 

16.648 

33.296 

.135 

.166 


20 

297.77 

1.857 

3.177 

3.965 

4.228 

16.792 

33.533 

.136 

.163 


30 

295.25 

1.873 

3.204 

4.000 

4.265 

16.935 

33.870 

.137 

.169 


40 

292.77 

1.890 

3.232 

4.034 

4.301 

17.078 

34.157 

.133 

.171 


50 

290.33 

1.906 

3.259 

4.069 

4.333 

17.222 

34.443 

.140 

.172 

20 

0 

237.94 

1.922 

3.2S7 

4.103 

4.374 

17.365 

34.730 

.141 

.174 




































TABLE II. LONG CHORDS 


119 


TABLE II. 

LONG CHORDS. § 69. 


Degree of 
Curve. 

2 Stations. 

3 Stations. 

4 Stations. 

5 Stations. 

6 Stations. 

o / 

0 10 

200.000 

299.999 

399.993 

499.996 

599.993 

20 

199.999 

.997 

.992 

.933 

.970 

30 

.998 

.992 

.981 

.962 

.933 

40 

.997 

.986 

.966 

.932 

.8S2 

50 

.995 

.979 

.947 

.894 

.815 

1 0 

199.992 

299.970 

399.924 

499.843 

599.733 

10 

.990 

.959 

.896 

.793 

.637 

20 

.9S6 

.946 

.865 

.729 

.526 

' 30 

.983 

.932 

.829 

.657 

.401 

40 

.979 

.915 

.789 

.577 

.260 

50 

.974 

.898 

.744 

.488 

.105 

2 0 

199.970 

299.878 

399.695 

499.391 

598.934 

10 

.964 

.857 

.643 

.285 

.750 

' 20 

.959 

.834 

.586 

.171 

.550 

30 

.952 

.810 

.524 

•049 

.336 

40 

.946 

.783 

.459 

49S.918 

.106 

50 

.939 

.756 

.389 

.778 

597.862 

3 0 

199.931 

299.726 

399.315 

498.630 

597.604 

10 

.924 

.695 

.237 

.474 

.331 

20 

.915 

.662 

.154 

.309 

.043 

30 

.907 

.627 

.068 

.136 

596.740 

40 

•89S 

.591 

398.977 

497.955 

.423 

50 

.888 

.553 

.832 

.765 

.091 

4 0 

199.878 

299.513 

398.782 

497.566 

595.744 

10 

.863 

.471 

.679 

.360 

.383 

20 

.857 

.423 

.571 

.145 

.007 

30 

.846 

.333 

.459 

496.921 

594.617 

40 

.834 

.337 

.343 

.689 

.212 

50 

.822 

.289 

.223 

.449 

593.792 

5 0 

199.810 

299.239 

393.099 

496.200 

593.358 

10 

.797 

.187 

397.970 

495.944 

592.909 

20 

.783 

.134 

.837 

.678 

446 

30 

.770 

.079 

.700 

.405 

591.963 

40 

.756 

.023 

.559 

.123 

.476 

50 

.741 

298.964 

.413 

494.832 

590.970 

6 0 

199.726 

298.904 

397.264 

494.534 

590.449 

10 

.710 

.843 

.110 

.227 

589.913 

20 

.695 

.779 

396.952 

493.912 

.364 

30 

.678 

.714 

.790 

.588 

588.800 

40 

.662 

.648 

.623 

257 

.221 

50 

. .644 

.579 

453 

492.917 

587.628 

7 0 

199.627 

298.509 

396.278 

492.563 

587.021 

10 

.609 

438 

099 

.212 

586.400 

20 

.591 

.364 

395.916 

491.847 

585.765 

30 

.572 

.289 

.729 

.474 

.115 

40 

•553 

.212 

.538 

.093 

584.451 

50 

.533 

.134 

.342 

490.704 

583.773 

8 0 

.513 

298.054 

395.142 

490.306 

583.081 




















120 


TABLE III 


TABLE IV, 


TABLE III. 

CORRECTION FOR THE EARTH’S CURVATURE AND 
FOR REERACTION. § 105. 


D. 

d. 

D. 

d. 

D. 

d. 

D. 

d. 

\ 

300 

.002 

1800 

.066 

3300 

.223 

4800 

A72 

400 

.003 

1900 

.074 

3400 

.237 

4900 

.492 

500 

.005 

2000 

.082 

3500 

.251 

5000 

.512 

600 

.007 

2100 

.090 

3600 

.266 

5100 

.533 

700 

.010 

2200 

.099 

3700 

.281 

5200 

.554 

800 

.013 

2300 

.108 

3800 

.296 

1 mile. 

.571 

900 

.017 

2400 

.118 

3900 

.312 

2 “ 

2.285 

1000 

.020 

2500 

.128 

4000 

.328 

3 “ 

5.142 

1100 

.025 

2600 

.139 

4100 

.345 

4 “ 

9.142 

J200 

.030 

2700 

.149 

4200 

.362 

5 « 

14.284 

5.300 

.035 

2800 

.161 

4300 

.379 

6 “ 

20.568 

1400 

.040 

2900 

.172 

4400 

.397 

7 “ 

27.996 

1500 

.046 

3000 

.184 

4500 

.415 

8 “ 

36.566 

1600 

.052 

3100 

.197 

4600 

.434 

9 “ 

46.279 

1700 

.059 

3200 

.210 

4700 

.453 

10 “ 

57.135 


TABLE IV. 

ELEVATION OF THE OUTER RAIL ON CURVES. 

§ 110 . 


Degree. 

* 

II 

Si 

M = 20. 

S3 

II 

8 

M = 30. 

s 

II 

M= 50. 

o 

1 

.012 

.022 

,034 

.049 

.088 

.137 

2 

.025 

.044 

.068 

.099 

.175 

.274 

3 

.037 

.066 

.103 

.148 

.263 

.411 

4 

.049 

.088 

.137 

.197 

.351 

.548 

5 

.062 

.110 

.171 

.247 

.438 

.685 

6 

.074 

.131 

.205 

.296 

.526 

.822 

7 

.086 

.153 

.240 

.345 

.613 

.958 

8 

.099 

.175 

.274 

.394 

.701 

1.095 

9 

.111 

.197 

.308 

.443 

.788 

1.232 

10 

.123 

.219 

.342 

.493 

.876 

1.368 












































121 



TABLE V. — TABLE 


TAB 


FROG ANGLES, CHORDS, 

TURNOUT 


This tabic is calculated fox' g = 4.7, d — .42, and S = 1° 20'. For 
mula for frog angle F, and chord B F, § 50; for m, the middle cm* 
tlinate of B F : § 25 ; for m 1 , the middle ordinate for cui'ving an 18 ft 
rail, § 29. 



R. 

F. 

BF. 

m. 

m'. 

R. 

F. 

BF. 

m. 


1000 

O II 

5 27 44 

72.22 

.651 

.041 

600 

O / II 

6 57 48 

59.17 

.727 

.068 

975 

5 31 39 

71.53 

.655 

.042 

575 

7 6 26 

58.16 

.733 

.070 

950 

5 35 44 

70.83 

.659 

.043 

550 

7 15 40 

57.12 

.739 

.074 

925 

5 39 59 

70.11 

.663 

.044 

525 

7 25 33 

56.05 

.745 

.077 

900 

5 44 24 

69.33 

.667 

.0-15 

500 

7 36 10 

54.94 

.752 

.031 

875 

5 49 1 

63.64 

.671 

.046 

475 

7 47 37 

53.79 

.758 

.085 

850 

5 53 50 

67.88 

.676 

.048 

450 

8 0 1 

52.61 

.765 

.090 

825 

5 58 52 

67.10 

.630 

.049 

425 

8 13 30 

51.37 

.773 

.095 

800 

6 4 9 

66.30 

.685 

.051 

400 

8 23 14 

50.09 

.780 

.101 

775 

6 9 41 

65.49 

.690 

.052 

375 

8 44 26 

48.75 

.788 

.108 

750 

6 15 30 

64.65 

.695 

.054 

350 

9 2 20 

47.35 

.796 

.116 

725 

6 21 37 

63.80 

.700 

.056 

325 

9 22 16 

45.88 

.805 

.125 

700 

6 28 4 

62.92 

.705 

.058 

300 

9 44 39 

44.34 

.814 

.135 

675 

6 34 52 

62.02 

.710 

.060 

275 

10 10 1 

42.72 

.824 

.147 

650 

6 42 4 

61.09 

.716 

.062 

250 

10 39 6 

41.00 

.834 

.162 

625 

6 49 42 

60.14 

.721 

.065 

225 

11 12 55 

39.16 

.845 

.180 


TABLE VI. 


LENGTH OF CIRCULAR ARCS IN PARTS OF RADIUS. 


o 

1 

.01745 

32925 

19943 

/ 

1 

\ 

.00029 

08882 

08666 

n 

1 

.00000 

48481 

36311 

2 

.03490 

65850 

39887 

2 

.00058 

17764 

17331 

2 

.00000 

96962 

73622 

3 

.05235 

98775 

59830 

3 

.00087 

26646 

25997 

3 

.00001 

45444 

10433 

4 

.06931 

31700 

79773 

4 

.00116 

35528 

34663 

4 

.00001 

93925 

47244 

5 

.08726 

64625 

99716 

5 

.00145 

44410 

43329 

5 

.00002 

42406 

84055 

6 

.10471 

97551 

19660 

6 

.00174 

53292 

51994 

6 

.00002 

90883 

20367 

7 

.12217 

30476 

39603 

7 

.00203 

62174 

60660 

7 

.00003 

39369 

57678 

8 

.13962 

63401 

59546 

8 

.00232 

71056 

69326 

8 

.00003 

87850 

94489 

9 

.15707 

96326 

79190 

9 

.00261 

79933 

77991 

9 

.00004 

36332 

31300 






































122 


TABLE VII. EXPANSION BY HEAT, 


TABLE VII. 

EXPANSION BY HEAT. 


Bodies. 

32^ to 2120. 

10. 

Authority. 

Platina, 

.0008,342 

.000004912 

Hassler 

Gold, 

.001466 

.000008141 

44 

Silver, 

.001909 

.000010605 

44 

Mercury, 

.018018 

.0001001 

(4 

Brass, 

.00189163 

.000010509 

44 

Iron, 

.00125344 

.000006964 

44 

Water, 

.0466 

not uniform. 

44 

Granite, 

.00036850 

.000004825 

Prof. Bartlett. 

Marble, 

.00102024 

.000005668 

44 

Sandstone, 

.00171576 

.000009532 

44 



















TABLE VIII. PROPERTIES OF MATERIALS. 


123 


TABLE VIII. 


PROPERTIES OF MATERIALS. 


The authorities referred to by the capital letters in the table are: — 


B Barlow, On the Strength of 
Materials. 

Be. Bevan. 

Br, Lieut. Brown. 

C. Couch. 

F. Franklin Institute, Report on 
Steam Boilers. 

6. Gordon, Eng. Translation of 
Weisbach. 

H. Hodgkinson, Reports to Brit. 

Association. 

Ha. Hassler, Tables. 


L. Lame. 

M- Musschenbroek, hit. to Nat 
Phil. 

R. Rennie, Phil. Trans. 

Ro. Rondelet, L'Art de Batir. 

T. Telford. 

Ta. Taylor, Statistics of Coal. 
W. Weisbach, Mech. of Machin¬ 
ery and Engineering. 

The numbers without letters are 
taken from Prof. Moseley’s En¬ 
gineering and Architecture 


In finding the weights, a cubic foot of water has, for convenience, 
been taken at 62.5 lbs. 

The numbers for compression taken from Hodgkinson were ob- 
tained by him from prisms high enough to allow the wedge of rupture 
to slide freely off. He shows that this is essential in experiments on 
compression. 

The modulus of rupture S is the breaking weight of a prism 1 in 
broad, 1 in. deep, and 1 in. between the supports, the weight being ap¬ 
plied in the middle. To find the corresponding breaking weight IF of 
a rectangular beam of any other size, let l = its length, b = its breadth, 

and cl = its depth, all in inches. Then W — X & 

The numbers in the last three columns express absolute strength 
For safety, a certain proportion only of these numbers is taken. Tho 
divisors for wood may be from 6 to 10, for metal from 3 to 6, for stone 
10, and for ropes 3. 

When double numbers are used in the column headed “ Crushing 
Force per Square Inch in lbs.,” the first applies to specimens moder¬ 
ately dry, the second to specimens turned and kept dry in a warm 
place two months longer. In the case of American Birch, Elm, and 
Teat, the numbers apply to seasoned specimens. 



124 


TABLE VIII. PROPERTIES OF MATERIALS, 




Weight 

Tensile 

Crushing 

3Iodulu8 

Materials. 

Specific 

Gravity. 

per 
Cubic 
Foot 
in lbs. 

Strength 
per Square 
Inch in lbs. 

Force per 
Square 
Inch 
in lbs. 

of Rup¬ 
ture S 
in lbs. 

• Metals. 

cast, . . ... 

8.399 

524.94 

17963 R. 



Gcppe.% oast,. 

8.607 

537.94 

19072 



“ rolled, .... 

8.864 F. 

554.00 

32826 F. 



u wire-drawn,. . . 

Gold,.j 

8.878 

554.87 

61228 



19.258 Ha. 

1203.62 




19.361 Ha. 

1210.06 




Iron, cast, 

Canon No. 2, cold blast, 

7.066 H. 

441.62 

16683 II. 

106375 IT. 

38556 II. 

“ “ hot “ 

7.046 II. 

440.37 

13505 H. 

108540II. 

37503 II. 

Devon No. 3, cold “ 

7.295 II. 

455.94 



36288 II. 

“ ! ‘ hot “ 

7.229 II. 

451/81 

21907 II. 

145435 H. 

43497 II. 

Buffery No. 1, cel® “ 

7.079 II. 

442.44 

17466 II. 

93335II. 

37503 II. 

“ “ hot “ 

6.998 II. 

437.37 

13434 H. 

86397 H. 

35316 H. 

Iron, wrought, 






English bar, . . 

7.700 

481.25 

57120 L. 

56000 ?G. 

54000 G. 

Welsh “ 



64960 T. 



Swedish “ .... 



64960 T. 

► 


U U 

7.478 F. 

467.37 

58184 F. 



Lancaster Co., A ■. orw, . 

7.740 F. 

483.75 

58661 F. 



Tennessee “ 

7.S05 F. 

487.81 

52099 F. 



Missouri u 

Iron wire, 

7.722 F. 

482 62 

47909 F. 



English, a'a.n .lfTK 



80214 T. 



Phillipsb’g, I'a. “ S33 u 

7.727 F. 

4S2.94 

84186 F. 



“ ‘ .i9v.’ " 



73888 F. 



“ “ .K« “ 



89162 F. 



Lead, cast,. 

11.446 M. 

715.37 

1824 R. 



Lead wire,. 

1 1 .IT 7 

707.31 

2531 31. 



Mercury,. 

.13.598 W. 

849.87 




Platina,. $ 

4 

1.1500 Ha. 
22.669 Ha. 

1218.75 

1416.81 




Silver,. 

10 474 HA. 

654.62 

40902 M. 



Steel, soft,. 

7.7S0 

486.25 

120000 



“ razor-temper"®, . . 

7.840 

490.90 

150000 



Tin, cast,. 


456.69 

6322 31. 



Zinc, fused, .. 

7 050 IV. 

44012 




“ rolled,. 


471.25 




Wo 0(tj. 




18683H.) 

| 9363II.) 


Ash, English,. 

.760 B 

47.50 

'/coon. 

12156 B 

Birch, English, .... 

.792 B. 

40.50 

CM, v) 

13297II. 1 
i 6402II.) 

10920 B. 

“ American,.... 

.648 B. 

40.50* 

1166311.' 

9624 B. 

Box,. 

.960 B. 

60.00 

2CC0G 

9771 II. 


Cedar, Canadian, . . 

909 C. 

56.81 

11400 Bo. 

<W<* II. 

1596" 71. 


Chestnut, .... 

.657 Ito. 

41.06 

13300 Ito. 


Deal, Christiania mi<? P», 

.698 B. 

43.62 

12400 


9364 B. 

“ Memel “ 

.590 B. 

36.87 



103S6 B. 

“ Norway Spruce, 

.340 

21.25, 

17600 



11 English,. 

.470 

29.37 

7000 



Elm, seasoned, .... 

.553 B 

34.56 

13489 31. 

10331IA 

5078 B. 

Fir, New England, . . . 

.553 B. 

34.56 


(5748 r 1 
\65S6U.) 

6612 R. 

“ Riga, . . . 

.753 B. 

47.06 

12000 B. 

66 V **. 

Lignum-vitaa, . 

1.220 

76.25 

11800 31. 


Mahogany, Spanish, . . 

i 

.son 

50.00 

16500 

1819SF. 

)8193 11. 

."IV 1 d 



















































TABLE VIII 


PROPERTIES OF MATERIALS 


1‘25 




Weight 

Tensile 

Crushing 
Force per 
Square 
Inch 
in lbs. 

- —0 

Modulus 

Materials. 

Specific 

Gravity. 

per 
Cubic 
Foot 
in lbs. 

Strength 
per Square 
Inch in lbs. 

of Rup¬ 
ture 5 
in lbs. 

Woods. 




t 


Oak, English,. 

.9316. 

5S.37 

10000 B. 


6484 H. 1 

1005SIIJ 

10032 B. 

“ Canadian, .... 

.872 B. 

54.50 

10253 

1 

4231 II.) 

5982 H.J 

10596 B. 

Pine, pitch, 

.660 B. 

41.25 

7818 M. 

1 

6790II.) 
6790II. j 

9792 B. 

“ red, . .... 

.657 B. 

41.06 



5395 H.1 
7518H.i 

8046 B. 

“ American, white, 

.455 Br. 

23.44 



7829 Br. 

“ “ Southern, 

.872 Br. 

54.50 



\ 3107II. 
5124 II. 

13987 Br. 

Poplar,. 

.333 M. 

23.94 

7200 Be. 



Teak,. 


46.56 

15000 B. 

12101 II. 

14772 B. 

Other Materials. 






Brick, red,. 

2.168 R. 

135.50 

281) 

803 R. 

340 W. 

u pale red. . . 

2.085 It. 

130.31 

300 

562 R. 

ISO W. 

Chalk,. j 

2.784 

1.869 

174.00 

116.81 


501 It. 


Coal, Penn, anthracite, . j 

1.327 Ta. 

82.94 




1.700 Ta. 

J 06.25 




“ “ semi-bituminous, 

1.453 Ta. 

90.81 




“ Md. “ 

1.552 Ta. 

97.00 




“ Penn, bituminous, . 

1.312 Ta. 

82.00 




“ Ohio “ 

1.270 Ta. 

79.37 




“ English “ 

Earth, 

1.259 Ta. 

78.69 




loamy hard-stamped, fresh, 

2.060 W. 

123.75 




“ “ dry, 

1.930 W. 

120.62 




garden, fresh, .... 

2.053 W. 

128.12 




“ dry, .... 

1.630 W. 

101.87 




dry, poor,. 

1.340 W. 

83.75 




Glass, plate,. 

2.453 

153.31 

9420 



Gravel,. 

1.920 

120.00 




Granite, Aberdeen, . . . 

2.625 R. 

164.06 


10914 R. 


Ivory, . 

1.826 

114.12 

16626 



Tiimp.strmn. . ) 

2.400 W. 

150.00 


1500 W, 

700 W. 

,. » 

2.860 W. 

178.75 


6000 W. 

1700 W 

Marble, white Italian, . . 

2.638 II. 

164.87 


9583 G. 

1062 

“ black Galway, . . 

2.695 H. 

16S.44 



2664 

Masonry, quarry stone, dry, 

2.400 W. 

150.00 




“ sandstone, “ 

2.050 W. 

128.12 




“ brick, dry, . j 

1.470 W. 
1.590 W. 

91.87 

99.37 




Ropes, 






hemp, under 1 inch diarn., 



92S0 W. 



“ from 1 to 3 in. “ 



7218 W. 



“ over 3 inches “ 



5156 W. 



Sand, river,. 

1.886 

117.87 





1.900 W. 

118.75 


1400 W. 

600 W. 


2.700 17. 

163.75 


13000 W. 

800 IV 

“ Dundee, .... 

2.530 It. 

158.12 


6630 It. 


“ Derby, red and friable, 

2.316 It. 

144.75 


3142 R. 


Slate, Welsh,. 

2.888 

180.50 

12300 



“ Scotch,. 



9600 
















































126 


TABLE IX. MAGNETIC VARIATION. 


TABLE IX. 

MAGNETIC VARIATION. 

TnE following table has been made up from various sources, prin¬ 
cipally, however, from the results of the United States Coast Survey, 
kindly furnished in manuscript by the Superintendent, Prof. A. D. 
Bache. “ These results,” he remarks in an accompanying note, “ are 
from preliminary computations, and may be somewhat changed by the 
final ones.” Among the other sources may be mentioned the Smith¬ 
sonian Contributions for 1852, Trans. Am. Phil. Soc. for 1846, Lond. 
Phil. Trans, for 1849, Silliman’s Journal for 1838, 1840, 1846, and 
1852, and the various American, British, and Russian Government 
Observations. The latitudes and longitudes here given are not always 
to be relied on as minutely correct. Many of them, for places in the 
Western States, were confessedly taken from, maps and other uncer¬ 
tain sources. Those of the Coast Survey Stations, however, as well 
as those of American and foreign Government Observatories and Sta¬ 
tions, are presumed to be accurate. 

It will be seen that the variation of the magnetic needle in the 
United States is in some places west and in others east. The line of no 
variation begins in the northwest part of Lake Huron, and runs through 
the middle of Lake Erie, the southwest corner of Pennsylvania, the 
central parts of Virginia, and through North Carolina to the coast. 
All places on the east of this line have the variation of the needle 
west, — all places on the west of this line have the variation of the 
needle east; and, as a general rule, the farther a place lies from this 
line, the greater is the variation. The position of the line of no varia¬ 
tion given above is the position assigned to it by Professor Loomis for 
the year 1840. But this line has for many years been moving slowly 
westward, and this motion still continues. Hence places whose varia¬ 
tion is west are every year farther and farther from this line, so that 
the variation west is constantly increasing. On the contrary, places 
whose variation is east are every year nearer and nearer to this line, 
so that the variation east is constantly decreasing. The rate of this 
increase or decrease, as the case may be, is said to average about 2' for 
the Southern States, 4' for the Middle and Western States, and 6' for 
the New England States.* The increase in Washington in 1840-2 
was 3 r 44.2in Toronto in 1841 - 2 it was 4' 46 2". The changes in 


* Prof Loomis in Silliman’s Journal. Tol. XXXIX., 1S4J. 



TABLE IX. MAGNETIC VARIATION. 


127 


Cambridge, Mass, maybe seen from the following determinations of the 
variation, taken from the Memoirs of the American Academy for 1846. 


Cambridge, 

U 

1708, 

o / 

9 0 

Cambridge, 1788, 

o * * 

6 38 

1742, 

8 0 

Boston, 1793, 

6 30 

iC 

1757, 

7 20 

Salem, 1805, 

5 57 

u 

1761, 

7 14 

“ 1808, 

5 20 

u 

1763, 

7 0 

“ 1810, 

6 22 

it 

1780, 

7 2 

Cambridge, 1810, 

7 30 

l( 

1782, 

6 46 

“ 1835, 

8 51 

i( 

1783, 

6 52 

“ 1840, 

9 18 


But besides this change in the variation, which may be called secu¬ 
lar, there is an annual and a diurnal change, and very frequently there 
are irregular changes of considerable amount. With respect to the 
annual change, the variation west in the Northern hemisphere is gen¬ 
erally found to be somewhat greater, and the variation east somewhat 
less, in the summer than in the winter months. The amount of this 
change is different in different places, but it is ordinarily too small to 
be of any practical importance. The diurnal change is well deter¬ 
mined. At Washington in 1840-2, the mean diurnal change in the 
variation was,* — 

Summer, 10 4.1 Autumn, 6 21.2 Winter, 5 9.1 Spring, 8 10.7 
At Toronto the means were, t — 



'1841. 

1843. 

1845. 

1847. 

1849. 

1850. 

1851. 

Winter, 

6.67 

5.64 

5.73 

7.28 

8.25 

8.01 

7.01 

Spring and Autumn, 

9.46 

9.36 

9.15 

10.08 

12.25 

10.90 

10.82 

Summer, 

12.38 

11.70 

13.36 

13.84 

14.80 

13.74 

12.61 


The diurnal change in the variation is such that the north end of the 
needle in the Northern hemisphere attains its extreme westerly posi¬ 
tion about 2 o’clock, P. M., and its extreme easterly position about 
8 o’clock, A. M. In places, therefore, whose variation is west, the 
maximum variation occurs about 2 P. M., while in places whose vari¬ 
ation is east, the maximum variation occurs about 8 A. M. In Wash¬ 
ington, according to the report of Lieutenant Gilliss, the maximum va¬ 
riation, taking the mean of two years’ observations, occurs at l h - 33 ln, 
P. M., the minimum at 8 h - 6 m - A. M. 

The determinations of the Coast Survey are distinguished by the 
letters C. S. attached to the name of the observer. In some instances 
the name of the nearest town has been added to the name of the Coast 
Survey station. 

* Lieut. Gilliss’s Report, Senate Document 172, 1845 

* London Philosophical Transactions. 1852 























128 


TABLE IX. MAGNETIC VARIATION 


Place. 

Lati¬ 

tude. 

Longi¬ 

tude. 

Authority. 

Date. 

Variation. 

Maine. 

o 

/ 

o 

/ 


Sept., 1817 

o 

i 

Agamenticus, 

43 

13.4 

70 

41.2 

T. J. Lee, C. S. 

10 

10.0 w . 

Bethel, 

Bowdoin Hill, Port- 

44 

28.0 

70 

51.0 

J. Locke, 

June, 1845 

11 

50.0 “ 

41.1 “ 

land, 

43 

38.8 

70 

16.2 

J. E. Hilgard, C. S. 

Aug., 1851 

11 

Cape Neddick,York 

43 

11.6 

70 

36.1 

J. E. Hilgard, C. S. 

Aug., 1851 

11 

9.0 “ 

Cape Small, 

43 

46.7 

69 

50.4 

G. W. Dean, C. S. 

Oct., 1851 

12 

5.5 “ 

Kennebunkport, 

43 

21.4 

70 

27.8 

J. E. Hilgard, C. S. 

Aug., 1851 

11 

23.6 “ 

Kittery Point, 

43 

4.8 

70 

43.3 

J. E. Hilgard, C. S. 

Sept., 1850 

10 

30.2 “ 

Mt. Pleasant, 

44 

1.6 

70 

49.0 

G. W. Dean, 0. S. 

Aug., 1851 

14 

32.0 “ 

Portland, 

43 

41'.0 

70 

20.5 

J. Locke, 

J. E. Hilgard, C. S. 

June, 1845 

11 

28.3 “ 

Richmond Island, 

43 

32.4 

70 

14.0 

Sept., 1S50 

12 

17.9 “ 

. New Hampshire. 
Fabyan’s Hotel, 

44 

16.0 

71 

29.0 

J. Locke, 

June, 1845 

11 

32.0 W. 

Hanover, 

43 

42.0 

72 

10.0 

Prof. Young, 

1839 

9 

15.0 “ 

Isle of Shoals, 

42 

59.2 

70 

36.5 

T. J. Lee, C. S. 

Aug., 1847 

10 

3.4 “ 

Patuccawa, 

43 

7.2 

71 

11.5 

G. W. Dean, C. S 

Aug., 1849 

10 

42.9 “ 

Unkonoonuc, 

42 

59.0 

71 

35.0 

J. S. Ruth, C. S. 

Oct., 1848 

9 

5.6 « 

Vermont. 









Burlington, 

44 

27.0 

73 

10.0 

J. Locke, 

June, 1845 

9 

22.0 W . 

Massachusetts. 









Annis-squam, 

42 

39.4 

70 

40.3 

G. W. Keely, C. S. 

Aug., 1849 

11 

36.7 W. 

Baker’s Island, 

42 

32.2 

70 

46.8 

G. W. Keely, C. S. 

Sept., 1849 

12 

17.0 “ 

Blue Ilill, Milton, 

42 

12.7 

71 

6.5 

T. J. Lee, C. S. { 

Sept, and ) 
Oct., 1845) 

9 

13.8 “ 

Cambridge, 

Chappaquidick,Ed- 

42 

22.9 

71 

7.2 

W. C. Bond, 

1852 

July, 1846 

10 

8.0 “ 

gartown, 

Coddon’s Hill, Mar- 

41 

22.7 

70 

28.7 

T. J. Lee, C. S. 

8 

47.7 “ 


blehead, 

42 

31.0 

70 

50.9 

G. W. Keely, C. S. 

Sept., 1849 

11 

49.8 “ 

Copecut Hill, 

41 

43.3 

71 

3.3 

T. J. Lee, C. S. { 

Sept, and I 
Oct., 1844) 

9 

12.1 “ 

Dorchester, 

42 

19.0 

71 

4.0 

W. C. Bond, 

1839 

9 

2.0 “ 

Fort Lee, Salem, 

42 

31.9 

70 

52.1 

G. W. Keely, C. S. 

Aug., 1849 

10 

14.5 « 

Ilyannis, 

41 

38.0 

70 

18.0 

T. J. Lee, C. S. 

Aug., 1846 

9 

22.0 “ 

Indian Ilill, 

41 

25.7 

70 

40.3 

T. J. Lee, C. S. 

Aug., 1846 

8 

49.3 “ 

Little Nahant, 

42 

26.2 

70 

55.5 

G. W. Keely, C. S. 

Aug., 1849 

9 

40.9 “ 

Nantasket, 

42 

18.2 

70 

54.0 

T. J. Lee, C. S. 

Sept., 1847 

9 

33.5 “ 

Nantucket, 

41 

17.0 

70 

6.0 

T. J. Lee, C. S. 

July, 1846 

9 

14.0 “ 

New Bedford, 

41 

38.0 

70 

54.0 

T. J. Lee, C. S. 

Oct., 1845 

8 

54.6 “ 

Shootflying Hill, 






Aug., 1846 



Barnstable, 

41 

41.1 

70 

20.5 

T. J. Lee, C. S. 

9 

40.1 “ 

Tarpaulin Cove, 

41 

28.1 

70 

45.1 

T. J. Lee, C. S. 

Aug., 1846 

9 

10.1 “ 

Rhode Island. 






Oct. and ) 
Nov.,1844 j 



Beacon-pole Hill, 

41 

59.7 

71 

26.7 

T. J Lee, C. S. j 

9 

29.8 W. 

McSparran Hill, 

41 

29.7 

71 

27.1 

T. J. Lee, C. S. 

July, 1844 

8 

53.3 “ ! 

Point Judith, 

41 

21.9 

71 

28.9 

11. II. Fauntleroy,C.S. 

Sept, 1847 

8 

59.4 “ j 

Spencer Hill, 

41 

40.7 

71 

29.3 

T. J. Lee, C. S j 

July and ) 
Aug. 1844) 

9 

11.9 “ 

Connecticut. 









Black Rock, Fair- 









field, 

41 

8.6 

73 

12.6 

J. Renwick, C. S. 

Sept., 1845 

6 

53.5 W. 

Bridgeport, 

4i 

10.0 

73 

11.0 

J. llenwiek, C. S. 

Sept., 1845 

6 

19.3 “ 

Fort Wooster, 
Groton Point, New 

41 

16.9 

72 

53.2 

J. S. Ruth, C. S. 

Aug., 1848 

7 

26.4 “ 

London, 

41 

18.0 

72 

0.0 

J. Renwick, C. S. 

Aug., 1845 

? 

29.5 “ 

iL -— 







1 






























TABLE IX. MAGNETIC VARIATION 


129 


Place. 

Lati¬ 

tude. 

Longi¬ 

tude. 

Authority. 

Date. 

Variation. 

Milford, 

o 

41 

16.0 

Q 

73 

1 

1.0 

J. Renwick, C. S. 

Sept., 1845 

o 

6 

38.3 W. 

New Haven, Pavil- 










ion, 

New Haven, Yale 

41 

18.5 

72 

55.4 

J. S. Ruth, C. S 

Aug., 1848 

6 

37.5 

LL 

College, 

41 

18.5 

72 

55.4 

J. Renwick, C. S. 

Sept., 1845 

6 

17.3 

u 

Norwalk, 

Oyster Point, New 

41 

7.1 

73 

24.2 

J. Renwick, C. S. 

Sept., 1844 

6 

46.3 

ll 

Haven, 

41 

17.0 

72 

55.4 

J. S. Ruth, C. S. 

Aug., 1848 

6 32.3 

u 

Sachem’s Head, 









Guilford, 

41 

17.0 

72 

43.0 

J. Renwick, C. S. 

Aug., 1845 

6 

15.2 

u 

Sawpits, 

40 

59.5 

73 

39.4 

J. Renwick, C. S. 

Sept., 1844 

6 

1.6 

LL 

Saybrook, 

41 

16.0 

72 

20.0 

J. Renwick, C. S. 

Aug., 1845 

6 49.9 

LL 

Stamford, 

41 

3.5 

73 

32.0 

J. Renwick, C. S. 

Sept., 1844 

8 

40.4 

u 

Stonington, 

41 

20.0 

71 

54.0 

J. Renwick, C. S. 

Aug., 1845 

7 38.2 

ll 

New York. 










Albany, 

42 

39.0 

73 

44.0 

Regents’ Report, 

1836 

6 

47.0 W. 

Bloomingdale Asy- 









lum, 

40 

48.8 

73 

57.4 

J. Locke, C. S. 

April, 1846 

5 

10 9 

ll 

Cole, Staten Island, 
Drowned Meadow, 

40 

31.8 

74 

13.8 

J. Locke, C. S. 

April, 1846 

5 

33.8 

u 

L. I., 

40 

56.1 

73 

3.5 

J. Renwick, C. S. 

Sept., 1845 

6 

3.6 

a 

Flatbush, L. I., 

40 

40.2 

73 

57.7 

J. Locke, C. S. 

April, 1846 

5 

54.6 

Li 

Greenport, L. I., 

41 

6.0 

72 

21.0 

J. Renwick, C. S. 

Aug., 1845 

7 

14.6 

ll 

Leggett, 

Lloyd's Harbor, 

40 

48.9 

73 

53.0 

It.II. Eauntleroy,C.S. 

Oct., 1847 

5 

40.6 

u 

L. I., 

40 

55.6 

73 

21.8 

J. Renwick, C. S. 

Sept., 1844 

6 

12.5 

Li 

New Rochelle, 

40 

52.5 

73 

47.0 

J. Renwick, C. S. 

Sept., 1844 

5 

31.5 

LL 

New York, 

40 

42.7 

74 

0.1 

J. Renwick, C. S. 

Sept., 1845 

6 

25.3 

LL 

Oyster Bay, L. I., 
Rouse’s Point, 

40 

52.3 

73 

31.3 

J. Renwick, C. S. 

Sept., 1844 

6 

53.6 

U 

45 

0.0 

73 

21.0 

Boundary Survey, 

Oct., 1845 

11 

28.0 

LL 

Sands Lighthouse, 








L.I., 

40 

51.9 

73 

43.5 

R.II. Fauntleroy,C.S. 

Oct., 1847 

6 

9.7 

Li 

Sands Point, L. I., 
Watchhill. Eire Isl- 

40 

52.0 

73 

43.0 

J. Renwick, C. S. 

Sept., 1845 

7 

14.6 

Li 

and, 

40 

41.4 

72 

58.9 

R.H. Fauntleroy,C.S. 

Oct., 1847 

7 

33 5 

a 

West Point, 

41 

25.0 

73 

56.0 

Prof. Davies, 

Sept., 1835 

6 

32.0 

a 

New Jersey. 










Cape May Light- 










house, 

3S 

55.8 

74 

57.6 

J. Locke, C. S. 

June, 1846 

3 

3.2 W. 

Chew, 

39 

48.2 

75 

9.7 

,J. Locke, 0. S. 

July, 1846 

3 20.4 

Li 

Church Landing, 

39 

40.9 

75 

30.3 

J. Locke, C. S. 

June, 1846 

*5 

45.8 

u 

Egg Island, 

39 

10.4 

75 

7.8 

J. Locke, C. S. 

June, 1846 

3 

18.2 

LL 

Hawkins, 

39 

25.5 

75 

17.1 

J. Locke, C. S. 

J. E. Ililgard, C. S. 

June, 1846 

2 58.7 

LL 

Mt.Itose,Prince ton, 

40 

22.2 

74 

42.9 

Aug., 1852 

5 

31.8 

Li 

Newark, 

40 

44.8 

74 

7.0 

J. Locke, C. S. 

April, 1846 

5 

32.7 

LL 

Pine Mountain, 
Port Noi’ris, 

39 

25.0 

75 

19 9 

J. Locke, C. S. 

June, 1846 

2 52.0 

a 

39 

14.5 

75 

1.0 

J. Locke, C. S. 

June, 1846 

3 

6.5 

u 

Sandy Hook, 

Town Bank, Cape 

40 

28.0 

73 

59.8 

J. Renwick, C. S 

Aug., 1844 

5 

54.0 

a 

May, 

33 

58.6 

74 

57.4 

J. Locke, C. S. 

June, 1846 

3 

3.2 

a 

Tucker’s Island, 

39 

30.8 

74 

16.9 

T. J. Lee, C. S. 

Nov., 1846 

4 

23.8 

LL 

White Hill, Bor- 










dentown, 

40 

8.3 

74 

43 8 

J. Locke, C S. 

April, IS46 

4 22.5 

u 

Pennsylvania. 
Girard College, 










Philadelphia, 

39 

58.4 

75 

9.9 

J. Locke, C. S. 

May, 1846 

3 50.7 W. 

Pittsburg, 

40 

26.0 

79 

58.0 

J. Locke, 

May, 1845 

0 33.1 

LL 

j Vauuxem, Bristol, j40 

5.9 

74 

52.7 

J. Locke, C. S. 

July, 1846 

4 20.5 

LL 


* Local attraction exists here, according to Prof. Locke. 

7 































130 


TABLE IX. MAGNETIC VARIATION. 


Place. 

Lati¬ 

tude. 

Longi¬ 

tude. 

Authority. 

Date. 

Variation. 

Delaware. 






Bombay Hook 
Lighthouse, 

O 1 

39 21.8 

O | 

75 30.3 

J. Locke, (J. S 

June, 1816 

O 1 

3 17.9 W 

Fort Delaware, Del- 






aware Diver, 

39 35.3 

75 33.8 

J. Locke, C. S. 

June, 1846 
July, 1846 

3 16.0 “ 

Lewes Landing, 

33 48.8 

75 11.5 

J. Locke, C. S. 

2 47.7 “ 

Pilot Town, 

38 47.1 

75 • 9.2 

J. Locke, C. S. 

July, 1846 

2 42.2 <( 

Sawyer, 

39 42.0 

75 33.5 

J. Locke, C. S. 

June, 1846 

2 47.8 « 

Wilmington, 

39 44.9 

75 33.6 

J. Locke, C. S. 

May, 1846 

2 31.8 “ 

Maryland. 






Annapolis, 

38 56.0 

76 35.0 

T. J. Lee, C. S. 

June, 1845 

2 14.0 W. 

Bodkin, 

39 8.0 

76 25.2 

T. J. Lee, C. S. 

April, 1847 

2 2.6 “ 

Finlay, 

39 24.4 

76 31.2 

J. Locke, C. S. 

April, 1846 

2 19.5 « 

Fort McHenry, 






Baltimore, 

39 15.7 

76 34.5 

T. J. Lee, C. S. 

April, 1847 

2 13.0 “ 

Hill, 

33 53.9 

76 52.5 

G. W. Dean, C. S. 

Sept., 1850 

2 15.4 “ 

Kent Island, 

39 1.8 

76 18.8 

J. Ileuston, C. S. 

July, 1849 

2 30.5 “ 

Marriott’s, 

33 52.4 

76 36.3 

T J. Lee, C. S. 

June, 1849 

2 5.2 “ 

North Point, 

39 11.7 

76 26.3 

T J. Lee. C. S. 

July, 1846 

1 42.1 “ 

Osborne’s Ituin, 

39 27.9 

76 16.6 

T J. Lee, C. S. 

June, 1845 

2 32.4 “ 

Poole’s Island, 

39 17.1 

76 15.5 

T J. Lee, C. S. 

June, 1847 

2 28.5 “ 

Rosanne, 

39 17.5 

76 42.8 

T. J. Lee, C. S. 

June, 1845 

2 12.0 “ 

Sopei*, 

39 5.1 

76 56.7 

G. W. Dean, C. S. 

July, 1850 

2 7.0 “ 

South Base, Kent 






j Island, 

33 53.8 

76 21.7 

T. J. Lee, C. S. 

June, 1845 

2 26.2 « 

SusqxxehannaLight- 
house, Ilavie de 





Grace, 

39 32.4 

76 4.8 

T J. Lee, C. S. 

July, 1847 

2 51.1 “ 

Taylor, 

33 59.8 

76 27.6 

T J. Lee, C. S. 

May, 1847 

2 18.4 “ 

Webb, 

39 5.4 

76 40.2 

G. W. Dean, C. S. 

Nov., 1850 

2 7.9 *' 

District of Colum¬ 
bia. 

Causten, George- 






town, 

33 55.5 

77 4.1 

G. W. Dean, C. S. 

June, 1851 

2 11.3 W, 

Washington, 

33 53.7 

77 2.8 

J. M. Gilliss, 

June, 1842 

1 26.0 “ 

Virginia. 

Charlottesville, 

33 2.0 

78 31.0 

Prof. Patterson, 

1835 

0 0.0 

Roslyn, Peters- 






burg, 

37 14.4 

77 23.5 

G. W. Dean, C. S. 

J. Locke, 

Aug., 1852 

0 26.4 W. 

Wheeling, 

40 8.0 

80 47.0 

April, 1845 

2 4.0 E. 

North Carolina. 






Bodie’s Island, 

35 47.5 

75 31.6 

C. 0. Boutelle, C. S. 

Dec., 1846 

1 13.4 W. 

Shellbank, 

38 3.3 

75 44.1 

C. O. Boutelle, C. S. 

Mar., 1847 
Feb., 1847 

1 44.8 “ 

Stevenson's Point, 

36 6.3 

76 11,0 

0 O. Boutelle, C. S. 

1 39.7 “ 

South Carolina. 






Breach Inlet, 

32 46.3 

79 48.7 

C. O. Boutelle. C. S. 

April, 1849 

2 16.5 E. 

Charleston, 

32 41.0 

79 53.0 

Capt. Barnett’ 

G. Davidson, C. S. 

May, 1841 

2 24.0 “ 

East Base, Edisto, 

32 33.3 

SO 10.0 

April, 1850 

2 53.6 “ 

Georgia. 






Athens, 

34 0.0 

S3 20.0 

Prof. McCay, 

1837 

4 31.0 E. 

i Columbus, 
Milledgeville, 

32 28.0 

35 10.0 

Geol. Survey, 

1839 

5 30.0 “ 

33 7.0 

33 20.0 

Geol. Survey, 

I. E. Ililgard, C. S. 

1838 

5 51.0 “ 

Savannah, j 32 5.0 

31 5.2 

April, 1S52 

3 45.0 “ 

- 






































TABLE IX. MAGNETIC VARIATION. 


131 


Place. j 

Lati¬ 

tude. 

Longi¬ 

tude. 

Authority. 

Date. 

Variation. 

Florida. 

O 1 

o / 



O 1 

Cape Florida, 

25 39.9 

80 9.4 

J. E. Ililgard, C S. 

Feb., 1850 

4 25.2 E. 

Cedar Keys, 

29 7.5 

83 2.8 J. E. Ililgard, C. S. 

Mar., 1852 

5 20.5 “ 

St. Marks Light, 

30 4.5 

84 12.5; 

J. E. Ililgard, C. S. 

April, 1S52 

5 29.2 “ 

Sand Key, 

24 27.2 

81 52.0 

J. E. Hilgard, C. S. 

Aug., 1849 

5 29.0 “ 

Alabama. 

Fort Morgan, Mo- 






bile Bay, 

30 13.8 

88 0.4 

R.II. Fauntleroy,C.S. 

May, 1817 

7 3.8 E. 

Tuscaloosa, 

Mississippi. 

33 12.0 

87 42.0 

Prof. Barnard, 

1839 

7 28.0 “ 

East Pascagoula, 

Texas. 

30 20.7 

88 31.4 

R.II. Fauntleroy,C.S. 

June, 1847 

7 12.4 E. 

Dollar Point, Gal- 






veston, 

29 26.0 

94 53.0 

R.II. Fauntleroy,C.S. 

April, 1848 

8 57.2 E. 

Mouth of Sabine, 

29 43.9 

93 51.5 

J. D. Graham, 

Feb., 1840 

8 40.2 “ 

Ohio. 






Carrolton, 

39 38.0 

84 9.0 

J. Locke, 

Sept., 1845 

4 45.4 E. 

Cincinnati, 

39 6.0 

84 22.0 

J. Locke, 

April, 1845 

4 4.0 “ 

Columbus, 

39 57.0 

83 3.0 

J. Locke, 

July, 1845 

2 29.3 “ 

Hudson, 

41 15.0 

81 26.0 

E. Loomis, 

1840 

0 52.0 « 

Marietta, 

39 26.0 

81 29.0 

J. Locke, 

April, 1845 

2 25.0 “ 

Oxford, 

39 30.0 

84 33.0 

J. Locke, 

Aug., 1845 

4 50.0 “ 

St. Mary’s, 

40 32.0 

84 19. C 

J. Locke, 

Sept., 1345 

3 4.0 “ 

Tennessee. 






Nashville, 

36 10.0 

86 49.0 

Prof. Hamilton, 

1835 

7 7.0 E. 

Michigan. 






Detroit, 

Indiana. 

42 24.0 

82 5S.0 

Geol. Report, 

1840 

2 0.0 E. 

Richmond, 

39 49.0 

S4 47.0 

J Locke, 

Sept., 1845 

4 52.0 E 

South Hanover, 

33 45.0 

85 23.0 

Prof. Dunn, 

1837 

4 35.0 “ 

Illinois. 






Alton, 

33 52.0 

90 12.0 

II. Loomis, 

1840 

7 45.0 E 

Missouri. 






St. Louis, 

Wisconsin. 

33 36.0 

89 36.0 

Col. Nicolla, 

1835 

8 49.0 E. 

Madison, 

43 5 0 

89 41.0 

U. S. Surveyors, 

Nov., 1839 

7 30.0 E. 

Prairie du Chien, 

43 1.0 

91 8.0 

U. S. Surveyors, 

Oct., 1839 

9 5.0 “ 

Iowa. 






Brown’s Settlement 

42 2.0 

91 18.0 

J. Locke, 

Sept., 1839 

9 4.0 E. 

Davenport, 

41 30.0 

90 34.0 

U. S. Surveyors, 

Sept., 1839 

7 50.0 “ 

Farmer’s Creek, 

42 13.0 

90 39.0 

J. Locke, 

Oct., 1839 

9 11.0 “ 

Wapsipinnicon 




Sept., 1839 


River, 

41 44.0 

90 39.0 

J. Locke, 

8 25.0 “ 

California. 






Point Conception, 

34 26.9 

120 26.0 

G. Davidson, 0. S. 

Sept., 1850 

13 49.5 E 


























132 


TABLE IX. MAGNETIC VARIATION 


Place. 

Lati¬ 

tude. 

Longi¬ 

tude. 

Authority. 

Date. 

Variation. 

Point Pinos, 
Monterey, 

O 1 

36 38.0 

O 1 

121 54.0 

G. Davidson, 0. S. 

Feb., 1851 

o / 

14 58.0 E. 

Presidio, San 






Francisco, 

37 47.8 

122 27.0 

G. Davidson, C. S. 

Feb., 1852 

15 26.9 « 

Sau Diego, 

32 42.0 

117 14.0 

G. Davidson, C. S. 

May, 1851 

12 29.0 “ 

Oregon. 
Cape Disap- 






pointment, 

46 16.6 

124 2.0 

G. Davidson, C. S. 

July, 1851 

20 45.0 E. 

Ewing Harbor, 

42 44.4 

124 21.0 

G. Davidson, C. S. 

Nov., 1851 

18 29.2 “ 

Washington 






Territory. 

Scarboro’ liar- 






bor, 

48 21.8 

124 37.2 

G. Davidson, C.S. 

Aug., 1852 

21 30.2 E. 

British Amer¬ 
ica. 






Montreal, 

45 30.0 

73 35.0 

Capt. Lefroy, 

1842 

8 58.0 W . 

Quebec, 

46 49.0 

71 16.0 

Capt. Lefroy, 

1842 

14 12.0 “ 

St. Johns, C. E. 

45 19.0 

73 13.0 

Capt. Lefroy, 

1842 

11 22.0 “ 

Stans tead, 

45 0.0 

72 13.0 

Boundary Survey, 

Nov., 1845 

11 33.0 “ 

Toronto, 

43 39.6 

79 21.5 

British Govern., 

Sept., 1S44 

1 27.2 

New Grenada 
P anama, 

8 57.2 

79 29.4 

W. II. Emory, 

Mar., 1349 

6 54.6 E. 

Eastern Hemi¬ 
sphere. 






Greenwich,Eng¬ 
land, 

51 23.0 

0 0.0 

Prof. Airy, 

1S41 

23 16.01V. 

Makers toun, 





Scotland, 

55 35.0 

2 31.0 VV. 

J. A. Broun, 

Paris Observatory 

1842 

25 23.6 “ | 

Paris, France, 

48 50.0 

2 20.0 E. 

Nov., 1851 

20 25.0 “ j 

Munich, Bava- 





ria. 

48 9.0 

11 37.0 “ 


1842 

16 43.0 « | 

St. Petersburg, 
Russia, 

59 56.0 

30 19.0 “ 

Russian Govern., 

1842 

6 21.1 “ 

Catherinenburg 






Siberia, 

56 51.0 

60 34.0 “ 

Russian Govern., 

1842 

6 33.9 E 

Nertchinsk, Si- 





beria, 

51 56.0 

116 31.0 “ 

Russian Govern., 

1842 

3 46.9 W. 

St. Helena, 

15 56.7 S. 

5 40.5 W. 

British Govern., 

Dec., 1845 

23 36.6 “ 

Cape of Good 
Hope, 

33 56.0 

18 2S.7 E. 

British Govern , 

Tuly, 1846 

29 8.0 “ 

Hobarton, Van 






Diemen’s Ld., 

42 52.5 • 

147 27.5 “ 

British Govern., 

Dec., 1848 

10 8.0 B. 

:—--J 





























TABLE X. TRIGONOMETRICAL FORMULAS. 


133 


TABLE X. 

TRIGONOMETRICAL AND MISCELLANEOUS FORMULAS 

Let A (fig. 57) be any acute angle, and let a perpendicular B (7 be 
drawn from any point in one side to the other side. Then, if the sides 



of the right triangle thus formed are denoted by letters, as in the fig 
ure, we shall have these six formulae: — 


1. 

A ° 

sin. A = — . 

4. 

cosec. A — ~ . 

2. 

. b 

cos. A = - . 

e 

5. 

sec. 


3. 

tan. A = r • 

0 

6. 

cot. 

A = i. 

a 


Solution of Right Triangles (fig. 57). 



Given. 

Sought. 


Formulae. 


7 

a, c 

A,B,b 

sin. A = * , cos. B — ~ c , 5= 

y(c + a) (c —a) 

8 

a, h 

A, B , c 

. a 

tan. A = £ , 

' n a 

cot. B = £ , 

c = «/a 2 + ^ s * 

9 

A, a 

B, b, c 

B = 90° — A, 

b = a cot. A, 

a 

C = slnTS * 

10 

A, b 

B , a, c 

B — 90° — A, 

a = b tan. A, 

b 

C cos. A ’ 

11 

A, c 

B, a, b 

B = 90° — A, 

a = c sin. A, 

b = c cos. A. 











134 


TABLE X. TRIGONOMETRICAL AND 


Solution of Oblique Triangles (fig. 58 ). 



12 

Given. 

A , B : a 

Sought. 

b 

b 

Formulae. 

a sin. B 

sin. A * 

13 

A, a, b 

B 

. _ b sin. A 

sin. B — — 0 — . 

14 

a, b, C 

A — B 

, , , t~>\ {a — b) can. i (A + B) 

tan. ^ [A B) — a + b 





rifs=J (a+ !> + <;), 

15 

a, b, c 

A 

- 

COS.M 



« 


2 N /.s (5 — a) (s — 6) (s — c) 

L sm. A — * 

K be 

16 

A, /?, G } a 

area 

«2 sin. B sin 0 

Area — 2 sin. A 

17 

18 

A, b , c 
a, b, c 

area 

area 

area = bbc sin. A. 

s=i(a + 6 + r M area=v's (s— a) (s — b) (e- e) 


General Trigpnometri'v*' Formula ?. 


19 

sin. 

2 A -f* cos. 2 A — 

1. 






20 

sin. 

(A ± B) = 

sin 

. A cos. 

B ± sin 

B cos. 

A. 


21 

cos 

(. A ±B) = 

cos. A cos. 

B if sin. 

A "hv 

B. 


22 

sin. 

2 A = 2 sin. 

A 

cos. A. 






23 

cos 

2 A = cos. 2 ^ 

4 - 

— sin. 2 x 

t = i — : 

2 sin 


4 • = 

3 cos.‘ 

24 

sin. 

2 A=±-± 

cos. 2 A. 






25 

cos 


cos. 2 A. 






26 

sin. 

A + sin. B 

— 

2 sin. ^ 

(A -j- B) 

cos. 

1 

2 

U 

B). 

27 

sin. 

A — sin. B 

= 

2 cos. £ 

(A + B) 

sin. 

1 

2 

(A 

B). 

28 

cos 

A -f- cos. B 

— 

2 cos. ^ 

(A + B) 

cos. 

1 

2 

(A- 

B). 

29 

cos 

B — cos. A 

- - 

2 sin. ^ 

(A + B) 

sin. 

1 

2 

(A- 

-B) 

30 

sin. 

2 A — sin. 2 B 


cos. 2 /? - 

- cos. 2 A ~ 

= sin. 

( 

A -f B) sf” 1 

31 

cos 

2 A — sin. 2 B 

- 

cos. [A 

B) cos 

(A 


-B). 
























MISCELLANEOUS FORMULAE 


1R5 


32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 


sin. A 


cos. A 
cos. A 
sin. A 


ian. A = 
cot. A = 
tan. 

tan. A ± tan. B — 


, t , ^ tan. A i tan. B 

(M ± B) = r -p tan. a tan. B ' 
sin. (A ± B) 
cos. A cos. B ’ 

4 , . 7~j j_ sin. ( A ± B) 

cot. A ± cot. B = ± 

sin. A A- sin. B tan. £ (A + B) 
tan. £ [A — B) ' 

tan. ± {A + B). 


sin. 

sin. 

cos. 

sin. 


A — sin. B 
A -f- sin. B 


cos. 

sin. 

cos. 

sin. 


A + cos. B 
A + sin. B . . . 

B — cos. A = cot ‘ 2 (A — 

A — sin. B , , . 

A + cos. B = tan< 2 B)' 


sin. B 


cos. B — cos. A 

sin. A 

tan. i A = i + coa ; 7 . 

. sin. A 

Q0L i A = 1 -cis. A ' 


cot. j (A + B). 


Miscellaneous Formulas. 



Sought. 

Area of 

Given. 

Formal®. 

44 

Circle 

Radius = r 

rt r*. 

45 

Ellipse 

Semi-axes = a and b 

n a b. 

46 

Parabola 

Chord = c, height — h 

%ch.* 

47 

Regular Polygon 

Surface of 

( Side = a, number of ) 

| sides — n ) 

, , 180° 

\arn cot. ——■•• 

48 

Sphere 

Radius = r 

4 n r 2 . 

49 

Zone 

Radius = r, height — h 

2 t i rh. 

50 

Spherical Polygon 

( Radius of sphere=r 1 
< sum of angles = S > 

( number of sides — n) 

9 <S -(n-2)180o 

711 * 180 3 


Solidity of 



'51 

Prism or Cylinder 

Base = b, height — h 

b h. 

52 

Pyramid or Cone 

Base = b, height == h 

$bh. 

53 

Frustum of Pyr- ) ( Bases = b and b t , 1 

amid or Cone ) | \ height — h ( 

£ h (b -f- b l + y/b b x ) 


* The area of a circular segment on railroad curves, where the chord is very long 
in proportion to the height, may he found with great accuracy hy the above formula. 


























?36 


TA.BLE X. MISCELLANEOUS FORMULAE, 



Soughs,. 
Solidity of 

54 

Sphere 

55 

SphericalSegment 

56 

Prolate Spheroid 

57 

Oblate Spheroid 

58 

Paraboloid 


Given. 


Radi ns = r 

( Radii of bases — r ) 
| and r l , height = h ) 

' Semi-transverse axis' 
of ellipse = a 
Semi-conjugate axis 
of ellipse = b 

( Radius of base = \ 

l height = h ) 


Formulae. 

| n r 3 . 

rA(r 2 4-r 1 2 -f§^*). 

17i ab 2 . 

5 n a 2 b. 

\7zr 2 h. 


tv = 3.14159 26535 89793 23846 26433 83280. 

Log. Ti = 0.49714 98726 94133 85435 12682 88291 

United States Standard Gallon = 231 cub. in. = 0.133681 cub.ft 
“ « « Bushel = 2150.42 “ = 1.244456 “ 

British Imperial Gallon = 277.27384 “ = 0.160459 “ 


French Metre, 

“ Litre, 

“ Kilogram, 


According to Ilassler. 

= 3.2817431 ft., 

= 61.0741569 cub. in., 
= 2.204737 lb. avoir., 


As usually given. 

= 3.280899 ft. 

= 61.02705 cub. in. 
= 2.2045971b. avoir 


Weight of Cubic Foot of Water, 

Barom. 30 inches, Therm. Fahr. 39.83°, 

n U I: 62° 


62.379 lb. avoir. 
62.321 “ 


Length of Seconds Pendulum at New York 
“ “ “ “ “ London 

“ “ “ “ « Paris 


= 39.10120 inches. 
= 39.13908 “ 

= 39.12843 “ 


Equatorial Radius of Earth according to Bessel 
Polar “ “ “ “ 


20,923,597.017 feet 
20,853,654.177 “ 


c( 








TABLE XI. 


SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, 


AND 


RECIPROCALS OF NUMBERS 


F&OM 1 TO 1054. 



{38 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.4142136 

1.2599210 

.500000000 

3 

9 

27 

1.7320508 

1.4422496 

.333333333 

4 

16 

64 

2.0000000 

1.5874011 

.250000000 

5 

25 

125 

2.2360630 

1.7099759 

.200000000 

6 

36 

216 

2.4494897 

1.8171206 

.166666667 

7 

49 

343 

2.6457513 

1.9129312 

.142857142 

8 

64 

512 

2.8234271 

2.0000000 

.125000000 

9 

81 

729 

3.0000000 

2.0300337 

.111111111 

10 

100 

1000 

31622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166248 

2.2239301 

.090909091 

12 

144 

1728 

3.4641016 

2.2894236 

.033333333 

13 

169 

2197 

3.6055513 

2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.4101422 

.071423571 

15 

225 

3375 

3.8729333 

2.4662121 

.066666667 

16 

256 

4096 

4.0000000 

2.5193421 

.062500000 

17 

239 

4913 

4.1231056 

2.5712316 

.058823529 

18 

324 

5832 

4.2426407 

2.6207414 

.055555556 

19 

361 

6359 

4.3588939 

2.6634016 

.052631579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

21 

441 

9261 

4.53257t>7 

2.7589243 

.047619048 

22 

434 

1064S 

4.6904153 

2.8020393 

.045454545 

23 

529 

12167 

4.7953315 

2.8438870 

.04347826 

24 

576 

13324 

4.8989795 

2.3844991 

.041666667 

25 

625 

15625 

5.0000000 

2.9240177 

.040000000 

26 

676 

17576 

5.0990195 

2.9524960 

.033461533 

27 

729 

19633 

5.1961524 

3.0000000 

.037037037 

28 

784 

21952 

5.2915026 

3.0365839 

.035714236 

29 

841 

21339 

5.3351648 

3.0723163 

.034482759 

30 

900 

27000 

5.4772256 

3.1072325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413306 

.032258065 

32 

1024 

32763 

5.6568542 

. 3.1743021 

.031250000 

33 

1039 

35937 

5.7445626 

3.2075343 

.030303030 

31 

1156 

39304 

5.8309519 

3.2396118 

.029411765 

35 

1225 

42375 

5.9160793 

3.2710663 

.028571429 

36 

1296 

46656 

6.0000000 

3.3019272 

.027777778 

37 

1369 

50653 

6.0827625 

3.3322218 

.027027027 

33 

1444 

54372 

6.1644140 

3.3619754 

.026315789 

39 

1521 

59319 

6.2449930 

3.3912114 

.025641026 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1631 

63921 

6.4031242 

3.4432172 

.024390244 

42 

1764 

740.33 

6.4S07407 

3.4760266 

.023809524 

43 

1349 

79507 

6.5574335 

3.5033981 

.023255814 

44 

1936 

85184 

6.6332496 

3.5303483 

.022727273 

45 

2025 

91125 

6.7032039 

3.5563933 

.022222222 

46 

2116 

97336 

6.7823300 

3.5330479 

.021739130 

47 

2209 

103323 

6.8556546 

3.6088261 

.021276600 

43 

2304 

110592 

6.9282032 

3.63-12411 

.020333333 

49 

2401 

117649 

7.0000000 

3.6593057 

.020403163 

50 

2500 

125000 

7.0710678 

3.6340314 

.020000000 

51 

2601 

132651 

7.141-4234 

3.7034298 

.019607843 

52 

2704 

140603 

7.2111026 

3.7325111 

.019230769 

53 

2309 

148377 

7.2301099 

3.7562358 

.018367925 

51 

2916 

157464 

7.3484692 

3.7797631 

.018518519 

55 

3025 

166375 

7.4161935 

3.8029525 

.018181818 

56 

3136 

175616 

7.4833148 

3.8258624 

.017857143 

57 

3249 

185193 

7.5498344 

3.8485011 

.017543860 

58 

3364 

195112 

7.6157731 

3.8703766 

.017241379 

59 

3481 

205379 

7.6311457 

3.8929965 

.016949153 

60 

3600 

216000 

7.7459667 

3.9148676 

.016666667 

61 

3721 

226931 

7.S 102497 

3.9364972 

.016393443 

62 

3344 

233323 

7.8740079 

3.9578915 

.016129032 
- - -— 
























CUBE ROOTS, AND RECIPROCALS. 139 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

63 

3969 

250047 

7.9372539 

3.9790571 

.015873016 

64 

4096 

262144 

8.0000000 

4.0000000 

.015625000 

65 

4225 

274625 

8.0622577 

4.0207256 

.015384615 

66 

4356 

287496 

8.1240384 

4.0412401 

.015151515 

67 

4489 

300763 

8.1853528 

4.0615480 

.014925373 

63 

4624 

314432 

8.2462113 

4.0816551 

.014705882 

69 

4761 

328509 

8.3066239 

4.1015661 

.014492754 

70 

4900 

343000 

8.3666003 

4.1212S53 

014285714 

71 

5041 

357911 

8.4261498 

4.1408178 

.014084507 

72 

5184 

373248 

8.4852814 

4.1601676 

.013888889 

73 

5329 

389017 

8.5440037 

4.1793390 

.013698630 

74 

5476 

405224 

8.6023253 

4.1983364 

.013513514 

75 

5625 

421875 

8.6602540 

4.2171633 

.013333333 

76 

5776 

438976 

8.7177979 

4.2358236 

.013157895 

77 

5929 

456533 

8.7749644 

4.2543210 

.012987013 

78 

6084 

474552 

8.8317609 

4.2726586 

.012820513 

79 

6241 

493039 

8.8881944 

4.2908404 

.012658228 

80 

6400 

512000 

8.9442719 

4.3088695 

.012500000 

81 

6561 

531441 

9.0000000 

4.3267487 

.012345679 

82 

6724 

551368 

9.0553851 

4.3444815 

.012195122 

83 

6889 

571787 

9.1104336 

4.3620707 

.012048193 

84 

7056 

592704 

9.1651514 

4.3795191 

.011904762 

85 

7225 

614125 

9.2195445 

4.3968296 

.011764706 

86 

7396 

636056 

9.2733185 

4.4140049 

.011627907 

S7 

7569 

658503 

9.3273791 

4.4310476 

.011494253 

88 

7744 

681472 

9.3808315 

4.4479602 

.011363636 

89 

7921 

704969 

9.4339811 

4.4647451 

.011235955 

90 

8100 

729000 

9.4868330 

4.4814047 

.011111111 

91 

8281 

753571 

9.5393920 

4.4979414 

.010989011 

92 

8464 

778688 

9.5916630 

4.5143574 

.010869565 

93 

8649 

804357 

9.6436508 

4.5306549 

.010752688 

94 

8836 

830584 

9.6953597 

4.5468359 

.010638298 

95 

9025 

857375 

9.7467943 

4.5629026 

.010526316 

96 

9216 

884736 

9.7979590 

4.5788570 

.010416667 

97 

9409 

912673 

9.8488578 

4.5947009 

.010309278 

98 

9804 

941192 

9.8994949 

4.6104363 

.010204082 

99 

9801 

970299 

9.9498744 

4.6260650 

.010101010 

100 

10000 

1000000 

10.0000000 

4.6415888 

.010000000 

101 

10201 

1030301 

10.0498756 

4.6570095 

.009900990 

102 

10404 

1061208 

10.0995049 

4.6723287 

.009803922 

103 

10609 

1092727 

10.1488916 

4.6875482 

.009708738 

104 

10816 

1124S64 

10.1980390 

4.7026694 

.009615385 

105 

11025 

1157625 

10.2469508 

4.7176940 

.009523810 

106 

11236 

1191016 

10.2956301 

4.7326235 

.009433962 

107 

11449 

1225043 

10.3440804 

4.7474594 

.009345794 

103 

11664 

1259712 

10.3923048 

4.7622032 

.009259259 

109 

11881 

1295029 

10.4403065 

4.7768562 

.009174312 

110 

12100 

1331000 

10.4880085 

4.7914199 

.009090909 

111 

12321 

1367631 

10.5356538 

4.8058955 

.009009009 

112 

12544 

1404928 

10.5830052 

4.8202845 

.008928571 

113 

12769 

1442897 

10.6301458 

4.8345881 

.008849558 

114 

12996 

-1481544 

10.6770783 

4.8488076 

.008771930 

115 

13225 

1520875 

10.7238053 

4.8629442 

.008695652 

116 

13456 

1560896 

10.7703296 

4.8769990 

.008620690 

117 

13689 

1601613 

10.8166538 

4.8909732 

.008547009 

118 

13924 

1643032 

10.8627805 

4.9048681 

.008474576 

119 

14161 

1685159 

10.9087121 

4.9186847 

.008403361 

120 

14400 

1728000 

10.9544512 

4.9324242 

.008333333 

121 

14641 

1771561 

11.0000000 

4.9460874 

.008264463 

122 

14884 

1815848 

11.0453610 

4.9596757 

.008196721 

123 

15129 

1860867 

11.0905365 

4.9731898 

.008130081 

124 

15376 

1906624 

11.1355287 

4.9866310 

.008064516 



















140 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

125 

15625 

1953125 

11.1803399 

5.0000000 

.003000000 

126 

15876 

2000376 

11.2249722 

5 0132979 

.007936508 

127 

16129 

2048333 

11.2694277 

5.0265257 

.007874016 

123 

16384 

2097152 

11.3137085 

5.0396842 

.007812500 

129 

16641 

2146639 

11.3578167 

5.0527743 

.007751938 

130 

16900 

2197000 

11.4017543 

5.0657970 

.007692308 

131 

17161 

2248091 

11.4455231 

5.0787531 

.007633588 

132 

17424 

2299963 

11.4891253 

5.0916434 

.007575758 

133 

17639 

2352637 

11.5325626 

5.1044637 

.007518797 

134 

17956 

2406104 

11.5758369 

5.1172299 

.007462687 

135 

18225 

2460375 

11.6189500 

5.1299278 

.007407407 

136 

18496 

2515456 

11.6619038 

5.1425632 

.007352941 

137 

18769 

2571353 

11.7046999- 

5.1551367 

.007299270 

133 

19044 

2623072 

11.7473444 

5.1676493 

.007246377 

139 

19321 

2635619 

11.7893261 

5.1801015 

.007194245 

140 

19600 

2744000 

11.8321596 

5.1924941 

.007142357 

141 

19831 

2303221 

11.8743421 

5.2048279 

.007092199 

142 

20164 

2363283 

11.9163753 

5.2171034 

.007042254 

143 

20449 

2924207 

11.9582607 

5.2293215 

.006993007 

144 

20736 

2935984 

12.0000000 

5.2414823 

.006944444 

145 

21025 

3048625 

12.0415946 

5.2535879 

.006396552 

146 

21316 

3112136 

12.0330460 

5.2656374 

.006849315 

147 

21609 

3176523 

12.1243557 

5.2776321 

.00680272. 

143 

21904 

3241792 

12.1655251 

5.2895725 

.006756757 

149 

22201 

3307949 

12.2065558 

5.3014592 

.006711409 

150 

22500 

3375000 

12 2474487 

5.3132923 

.006666667 

151 

22301 

3442951 

12.2882057 

5.3250740 

.006622517 

152 

23104 

3511808 

12.3238280 

5.3363033 

.006578947 

153 

23409 

3531577 

12.3693169 

5.3484812 

.006535948 

154 

23716 

3652264 

12.4096736 

5.3601084 

.006493506 

155 

24025 

3723375 

12.4498996 

5.3716354 

.006451613 

156 

24336 

3796416 

12.4399960 

5.3332126 

.006410256 

157 

24649 

3369393 

12.5299641 

5.3946907 

.006369427 

153 

24964 

3944312 

12.5693051 

5.4061202 

.006329114 

159 

25281 

4019679 

12.6095202 

5.4475015 

.006289308 

160 

25600 

4096000 

12.6491106 

5.4238352 

.006250000 

161 

25921 

4173281 

12.6335775 

5.4401218 

.006211180 

162 

26244 

4251528 

12.7279221 

5.4513618 

.006172340 

163 

26569 

4330747 

12.7671453 

5.4625556 

.006134969 

164 

26896 

4410944 

12.8062485 

5.4737037 

.006097561 

165 

27225 

4492125 

12.8452326 

5.4348066 

.006060606 

166 

27556 

4574296 

12.8340937 

5.4958647 

.006024096 

167 

27839 

4657463 

12.9228430 

5.5063784 

.005988024 

163 

28224 

4741632 

12.9614814 

5.5178434 

.005952381 

169 

23561 

4326309 

13.0000000 

5.5287748 

.005917160 

170 

23900 

4913000 

13.0334048 

5.5396583 

.005S82353 

171 

29241 

5000211 

13.0766968 

5.5504991 

.005847953 

172 

29534 

5088448 

13.1143770 

5.5612978 

.005813953 

173 

29929 

5177717 

13.1529464 

5.5720546 

.005780347 

174 

30276 

5263024 

13.1909060 

5.5827702 

.005747126 

175 

30625 

5359375 

13.2287566 

5.5934447 

.005714286 

176 

30976 

5451776 

13.2664992 

5.6040787 

.005631818 

177 

31329 

5545233 

13.3041347 

5.6146724 

.005649718 

178 

31684 

5639752 

13.3416641 

5.6252263 

.005617978 

179 

32041 

5735339 

13.3790332 - 

5.6357408 

.005586592 

180 

32400 

5832000 

13 4164079 

5.6462162 

.005555556 

131 

32761 

5929741 

13.4536240 

5.6566528 

.005524862 

182 

33124 

6028563 

13.4907376 

5.6670511 

005494505 

183 

33489 

6128487 

13.5277493 

5.6774114 

.005464481 

134 

33356 

6229504 

13.5646600 

5.6377340 

.005434783 

185 

34225 

6331625 

13.6014705 

5.6980192 

.005405405 

186 

34596 

0434856 

13.6331817 

5.7032675 

.005376344 






















CUBE ROOTS, AND RECIPROCALS 


141 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

187 

34969 

6539203 

13.6747943 

5.7184791 

.005347594 

188 

35344 

6644672 

13.7113092 

5.7286543 

.005319149 

189 

35721 

6751269 

13.7477271 

5.7387936 

.005291005 

190 

36100 

6859000 

13.7S40488 

5.7488971 

.005263158 

191 

36481 

6967871 

13.8202750 

5.7589652 

.005235602 

192 

36864 

7077888 

13.8564065 

5.76899S2 

.005208333 

193 

37249 

7189057 

13.8924440 

5.7789966 

.005181347 

194 

37636 

73013S4 

13.9283883 

5.7889604 

.005154639 

195 

38025 

7414875 

13.9642400 

5.7988900 

.005128205 

196 

38416 

7529536 

14.0000000 

5.8087857 

.005102041 ' 

197 

38809 

7645373 

14.0356688 

5.8186479 

.005076142 

198 

39204 

7762392 

14.0712473 

5.8284767 

.005050505 

199 

39601 

7880599 

14.1067360 

5.8382725 

005025126 

200 

40000 

8000000 

14.1421356 

5.8480355 

.005000000 

201 

40401 

8120601 

14.1774469 

5.8577660 

.004975124 

202 

40804 

8242408 

14.2126704 

5.8674643 

.004950495 

203 

41209 

8365427 

14.2478068 

5.8771307 

.004926108 

204 

41616 

8489664 

14.2828569 

5.8867653 

.004901961 

205 

42025 

8615125 

14.3178211 

5.8963685 

.004878049 

206 

42436 

8741816 

14.3527001 

5 9059406 

i .004854369 

207 

42849 

8869743 

14.3874946 

5.9154817 

.004830918 

208 

43264 

8998912 

14.4222051 

5.9249921 

.004807692 

209 

43681 

9129329 

14.4568323 

5.9344721 

.004784689 

210 

44100 

9261000 

14.4913767 

5.9439220 

.004761905 

211 

44521 

9393931 

14.5258390 

5.9533418 

.004739336 

212 

44944 

9528128 

14.5602198 

5.9627320 

.004716981 

213 

45369 

9663597 

14.5945195 

5.9720926 

.004694836 

214 

45796 

9800344 

14.6287388 

5.9814240 

.004672897 

215 

46225 

9938375 

14.6628783 

5.9907264 

.004651163 

216 

46656 

10077696 

14.6969385 

6.0000000 

.004629630 

217 

47089 

10218313 

14.7309199 

6.0092450 

.004608295 

218 

47524 

10360232 

14.7648231 

6.0184617 

.004587156 

219 

47961 

10503459 

14.7986488 

6.0276502 

.004566210 

220 

48400 

10648000 

14.8323970 

6.0368107 

.004545455 

221 

48841 

10793861 

14.8660687 

6.0459435 

.004524887 

222 

49284 

10941048 

14.8996644 

6 0550489 

.004504505 

223 

49729 

110S9567 

14.9331845 

6.0641270 

.004484305 

224 

50176 

11239424 

14.9666295 

6.0731779 

.004464286 

225 

50625 

11390625 

15.0000000 

6.0822020 

.004444444 

226 

51076 

11543176 

15.0332964 

6.0911994 

.004424779 

227 

51529 

11697083 

15.0665192 

6.1001702 

.004405286 

228 

51984 

11852352 

15.0996689 

6.1091147 

.004385965 

229 

52441 

12008989 

15.1327460 

6.1180332 

.004366812 

230 

52900 

12167000 

15.1657509 

6.1269257 

.004347826 

231 

53361 

12326391 

15.1986842 

6.1357924 

.004329004 

232 

53824 

12487168 

15.2315462 

6.1446337 

.004310345 

233 

54289 

12649337 

15.2643375 

6.1534495 

.004291845 

234 

54756 

12812904 

15.2970585 

6.1622401 

.004273504 

235 

55225 

12977875 

15.3297097 

6.1710058 

.004255319 

236 

55696 

13144256 

15.3622915 

6.1797466 

.004237288 

237 

56169 

13312053 

15.3948043 

6.1884628 

.004219409 

238 

56644 

13481272 

15.4272486 

6.1971544 

.004201681 

239 

57121 

13651919 

15.4596248 

6.2058218 

.004184100 

240 

57600 

13824000 

15.4919334 

6.2144650 

.004166667 

241 

58081 

13997521 

15.5241747 

6.2230843 

.004149378 

242 

58564 

14172488 

15.5563492 

6.2316797 

.004132231 

243 

59049 

14348907 

15.5884573 

6.2402515 

.004115226 

244 

59536 

14526784- 

15.6204994 

6.2487998 

.004098361 

245 

60025 

14706125 

15.6524758 

6.2573248 

.004081633 

246 

60516 

14886936 

15.6843871 

6.2658266 

.004065041 

247 

61009 

15069223 

15.7162336 

6.2743054 

.004048583 

248 

61504 

15252992 1 

15.7480157 

6.2827613 

.004032258 





















142 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

249 

62001 

15438249 

15.7797338 

6.2911946 

.004016064 

250 

62500 

15625000 

15.8113883 

6.2996053 

.004000000 

251 

63001 

15813251 

15.8429795 

6.3079935 

.003934064 

252 

63504 

16003003 

15.8745079 

6.3163596 

.003963254 

253 

64009 

16194277 

15.9059737 

6.3247035 

.003952569 

254 

64516 

16387064 

15.9373775 

6.3330256 

.003937008 

255 

65025 

16531375 

15.9637194 

6.3413257 

.003921569 

256 

65536 

16777216 

16.0000000 

6.3196042 

.003906250 

257 

66049 

16974593 

16.0312195 

6.3578611 

.003891051 

253 

66564 

17173512 

16.0623784 

6.3660968 

.003875969 

259 

67031 

17373979 

16.0934769 

6.3743111 

.003861004 

260 

67600 

17576000 

16.1245155 

6.3325043 

.003846154 

261 

63121 

17779581 

16.1554944 

6.3906765 

.003831413 

262 

63644. 

17984728 

16.1864141 

6.3938279 

.003816794 

263 

69169 

18191447 

16.2172747 

6.4069585 

.003302281 

264 

69696 

18399744 

16.2430768 

6.4150637 

.003787879 

265 

70225 

18609625 

16.2788206 

6.4231583 

.003773535. 

266 

70756 

18821096 

16.3095064 

6.4312276 

.003759393 

267 

71289 

19034163 

16.3401346 

6.4392767 

.003745318 

263 

71824 

19243332 

16.3707055 

6.4473057 

.003731343 

269 

72361 

19465109 

16.4012195 

6.4553148 

.003717472 

270 

72900 

19633000 

16.4316767 

6.4633041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003690037 

272 

73934 

20123643 

16.4924225 

6.4792236 

.003676471 

273 

74529 

20346417 

16.5227116 

6.4871541 

.003663004 

274 

75076 

20570324 

16.5529454 

6.4950653 

.003649635 

275 

75625 

20796375 

16.5331240 

6.5029572 

.003636364 

276 

76176 

21024576 

16.6132477 

6.5103300 

.003623188 

277 

76729 

21253933 

16.6433170 

6.5136839 

.003610108 

278 

77284 

21484952 

16.6733320 

6.5265189 

.003597122 

279 

77341 

21717639 

16.7032931 

6.5343351 

.003584229 

280 

78400 

21952000 

16.7332005 

6.5421326 

.003571429 

281 

78961 

22188041 

16.7630546 

6.5499116 

.003558719 

282 

79524 

22425763 

16.7928556 

6.5576722 

.003546099 

283 

80039 

22665187 

16.8226033 

6.5654144 

.003533569 

284 

80656 

22906304 

16.8522995 

6.5731335 

.003521127 

285 

81225 

23149125 

16.8819430 

6.5803443 

.003508772 

236 

81796 

23393656 

16.9115345 

6.5885323 

.003496503 

237 

82369 

23639903 

16.9410743 

6.5962023 

.003484321 

238 

82944 

23887872 

16.9705627 

6.6038545 

.003172222 

239 

83521 

24137569 

17.0000000 

6.6114390 

.003460208 

290 

84100 

24389000 

17.0293864 

6.6191060 

.003448276 

291 

84631 

24642171 

17.0537221 

6.6267054 

.003436426 

292 

85264 

24897038 

• 17.0880075 

6.6342874 

.003424658 

293 

85849 

25153757 

17.1172428 

6.6113522 

.003412969 

294 

86436 

25412184 

17.1464282 

6.6493998 

.003401361 

295 

87025 

25672375 

17.1755610 

6.6569302 

.003389831 

296 

87616 

25934336 

17.2046505 

6.6644437 

.003378378 

297 

83209 

26193073 

17.2336879 

6 6719403 

.003367003 

298 

83804 

26463592 

17.2626765 

6.6794200 

.003355705 

299 

89401 

26730399 

17.2916165 

6.6363831 

.003344482 

300 

90000 

27000000 

17.3205081 

6.6943295 

.003333333 

301 

90601 

27270901 

17.3493516 

6.7017593 

.003322259 

302 

91204 

27543603 

17.3781472 

6.7091729 

.003311258 ! 

303 

91809 

27818127 

17.4063952 

6.7165700 

.003300330 

304 

92416 

23094464 

17.4355958 

6.7239508 

.003289474 

305 

93025 

23372625 

17.4642492 

6.7313155 

.003278689 

306 

93636 

28652616 

17.4928557 

6.7336641 

.003267974 

307 

94249 

28934443 

17.5214155 

6.7459967 

.003257329 

303 

91364 

29218112 

17.5499288 

6.7533134 

.003246753 | 

3U9 

95431 

29503629 

17.5783958 

6.7606143 

.003236246 

310 

96100 

29791000 

17.6063169 

6.7678995 

.003225806 



































143 


CUBE ROOTS, AND RECIPROCALS. 


V* 

No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

311 

96721 

30080231 

17.6351921 

6.7751690 

.003215434 

312 

97344 

30371328 

17.6635217 

6.7824229 

.003205128 

313 

97969 

30664297 

17.6918060 

6.7896613 

.003194888 

314 

93596 

30959144 

17.7200451 

6.7968844 

.003184713 

315 

99225 

31255875 

17.7482393 

6.8040921 

.003174603 

316 

99356 

31554496 

17.7763888 

6.8112847 

.003164557 

317 

100469 

31855013 

17.804493S 

6.81S4620 

.003154574 

31S 

101124 

32157432 

17.8325545 

6.8256242 

.003144654 

319 

101761 

32461759 

17.8605711 

6.8327714 

.003134796 

320 

102400 

32768000 

17.8885438 

6.8399037 

.003125000 

321 

103041 

33076161 

17.9164729 

6.8470213 

.003115265 

322 

103634 

33386248 

17.9443584 

6.8541240 

.003105590 

323 

104329 

33698267 

17.9722008 

6.8612120 

.003095975 

324 

104976 

34012224 

18.0000000 

6.8682855 

.003086420 

325 

105625 

34328125 

18.0277564 

6.8753443 

.003076923 

326 

106276 

34645976 

18.0554701 

6.8823S88 

.003067485 

327 

106929 

34965783 

18.0831413 

6.8894188 

.003058104 

323 

107534 

35287552 

18.1107703 

6.8964345 

.003048780 

329 

108241 

35611289 

IS.1383571 

6.9034359 

.003039514 

330 

108900 

35937000 

18.1659021 

6.9104232 

.003030303 

331 

109561 

36264691 

18.1934054 

6.9173964 

.003021148 

332 

110224 

36594368 

18.2208672 

6.9243556 

.003012048 

333 

110869 

36926037 

18.2482876 

6.9313008 

.003003003 

334 

111556 

37259704 

18.2756669 

6.9382321 

.002994012 

335 

112225 

37595375 

18.3030052 

6.9451496 

.002985075 

336 

112896 

37933056 

18.3303028 

6.9520533 

.002976190 

337 

113569 

38272753 

18.3575598 

6.9589434 

.002967359 

333 

114244 

38614472 

18.3847763 

6.9658198 

.002958580 

339 

114921 

38958219 

18.4119526 

6.9726826 

:002949853 

340 

115600 

39304000 

18.4390889 

6.9795321 

.002941176 

341 

116231 

39651S21 

18.4661853 

6.9863681 

.002932551 

342 

116964 

40001688 

18.4932420 

6.9931906 

.002923977 

343 

117649 

40353807 

18.5202592 

’ 7.0000000 

.002915452 

344 

118336 

40707534 

18.5472370 

7.0067962 

.002906977 

345 

119025 

41063625 

18.5741756 

7.0135791 

.002898551 

346 

119716 

41421736 

18.6010752 

7.0203490 

.002890173 

347 

120409 

41781923 

18.6279380 

7.0271058 

.002881844 

343 

121104 

42144192 

18.6547581 

7.0338497 

.002873563 

349 

121801 

42508549 

18.6815417 

7.0405806 

.002S65330 

350 

122500 

42875000 

18.7082869 

7.0472987 

.002857143 

351 

123201 

43243551 

18.7349940 

7.0540041 

.002849003 

352 

123904 

43614208 

18.7616630 

7.0606967 

.002840909 

353 

124609 

43986977 

18.7882942 

7.0673767 

.002832861 

361 

125316 

44361864 

18.8148877 

7.0740440 

.002824859 

355 

126025 

41738875 

18.8414437 

7.0806988 

.002816901 

356 

126736 

45118016 

18.8679623 

7.0873411 

.002808989 

357 

127449 

45499293 

18.8944436 

7.0939709 

.002801120 

353 

128164 

45832712 

18.9208879 

7.1005885 

.002793298 

359 

128881 

462S8279 

18.9472953 

7.1071937 

.002785515 

360 

129600 

46656000 

18.9736660 

7.1137866 

.002777778 

361 

130321 

47045881 

19.0000000 

7.1203674 

.002770083 

362 

131044 

47437928 

19.0262976 

7.1269360 

.002762431 

363 

131769 

47832147 

19.0525589 

7.1334925 

.002754821 

364 

132496 

48228544 

19.0787840 

7.1400370 

.002747253 

365 

133225 

48627125 

19.1049732 

7.1465695 

.002739726 

366 

133956 

49027896 

19.1311265 

7.1530901 

.002732240 

367 

134639 

49430863 

19.1572441 

7.1595988 

.002724796 

363 

135424 

49836032 

19.1833261 

7.1660957 

.002717391 

369 

136161 

50243409 

19.2093727 

7.1725809 

.002710027 

370 

136900 

50653000 

19.2353841 

7.1790544 

.002702703 

371 

137641 

51064811 

19.2613603 | 

7.1855162 

.002695418 

372 

]383S4 

51478848 

19.2873015 I 

7.1919663 

.002688172 































144 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

373 

139129 

51895117 

19.3132079 

7.1984050 

.002680965 

374 

139376 

52313624 

19.3390796 

7.2048322 

.002673797 

375 

140625 

52734375 

19.3649167 

7.2112479 

.002666667 

376 

141376 

53157376 

19.3907194 

7.2176522 

.002659574 

377 

142129 

53532633 

19.4164878 

7.2240450 

.002652520 

378 

142384 

54010152 

19.4422221 

7.2304268 

.002645503 

379 

143641 

54439939 

19.4679223 

7.2367972 

.002638522 

330 

144400 

54872000 

19.4935887 

7.2431565 

.002631579 

331 

145161 

55306341 

19.5192213 

7.2495045 

.002624672 

382 

145924 

55742968 

19.5448203 

7.2558415 

.002617801 

333 

146639 

56181837 

19.5703858 

7.2621675 

.002610966 

334 

147456 

58623104 

19.5959179 

7.2684824 

.002604167 I 

335 

148225 

57066625 

19.6214169 

7.2747864 

.002597403 

336 

148996 

57512456 

19.6463327 

7.2310794 

.002590674 

337 

149769 

57960603 

19.6723156 

7.2873617 

.002583979 

333 

150544 

58411072 

19.6977456 

7.2936330 

.002577320 

339 

151321 

53363369 

19.7230829 

7.2998936 

.002570694 

390 

152100 

59319000 

19.7484177 

7.3061436 

.002564103 

391 

152331 

59776471 

19.7737199 

7.3123828 

.002557545 

392 

153664 

60236238 

19.7939899 

7.3186114 

.002551020 

393 

154449 

60698457 

19.8242276 

7.3248295 

.002544529 

394 

155236 

61162934 

19.8494332 

7.3310369 

.002538071 

395 

156025 

61629375 

19.8746069 

7.3372339 

.002531646 

396 

156316 

62099136 

19.8997437 

7.3434205 

.002525253 

397 

157609 

62570773 

19.9248538 

7.3495966 

.002518892 

393 

158404 

63044792 

19.9499373 

7.3557624 

.002512563 

399 

159201 

63521199 

19.9749844 

7.3619178 

.002506266 

400 

160000 

64000000 

20.0000000 

7.36S0630 

.002500000 

401 

160801 

64481201 

20.0249844 

7.3741979 

.002493766 

402 

161604 

64964803 

20.0499377 

7.3803227 

.002487562 

403 

162409 

65450827 

20.0748599 

7.3864373 

.002481390 

404 

163216 

65939264 

20.0997512 

7.3925418 

.002475248 

405 

164025 

66430125 

20.1246118 

7.3986363 

.002469136 

406 

161336 

66923416 

20.1494417 

7.4047206 

.002463054 

407 

165649 

67419143 

20.1742410 

7.4107950 

.002457002 

403 

166464 

67917312 

20.1990099 

7.4168595 

.002450980 

409 

167231 

63417929 

20.2237434 

7.4229142 

.0024449S8 

410 

163100 

63921000 

20.2484567 

7.4289589 

.002439024 

411 

163921 

69426531 

20.2731349 

7.4349938 

.002433090 

412 

169744 

69934523 

20.2977831 

7.4410189 

.002427184 

413 

170569 

70444997 

20.3224014 

7.4470342 

.002421308 

414 

171396 

70957944 

20.3469399 

7.4530399 

.002415459 

415 

172225 

71473375 

20.3715488 

7.4590359 

.002409639 

416 

173056 

71991296 

20.3960781 

7.4650223 

.002403846 

417 

173389 

72511713 

20.4205779 

7.4709991 

.002398082 

418 

174724 

73034632 

20.4450433 

7.4769664 

.002392344 

419 

175561 

73560059 

20.4694895 

7.4829242 

.002386635 

420 

176400 

740SS000 

20.4939015 

7.4888724 

.002380952 

421 

177241 

74618461 

20.5182345 

7.4943113 

.002375297 

422 

178034 

75151448 

20.5426336 

7.5007406 

.002369668 

423 

178929 

75636967 

20.5669633 

7.5066607 

.002364066 

424 

179776 

• 76225024 

20.5912603 

7.5125715 

.002358491 

425 

130625 

76765625 

20.6155281 

7.5184730 

.002352941 

426 

181476 

77308776 

20.6397674 

7.5243652 

.002347418 

427 

182329 

77854483 

20.6639783 

7.5302482 

.002341920 

423 

183184 

78402752 

20.6881609 

7.5361221 

.002336449 

429 

184041 

78953589 

20.7123152 

7.5419867 

.002331002 

430 

134900 

79507000 

20.7364414 

7.5478423 

.002325581 

431 

185761 

80062991 

20.7605395 

7.5536388 

.002320186 

432 

186624 

80621563 

20.7846097 

7.5595263 

.002314815 

433 

187489 

81182737 

20.8036520 

7.5653548 

.002309469 

434 

188356 

81746504 

20.8326667 

7.5711743 

.002304147 



















145 


CUBE ROOTS, AND RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

433 

189225 

82312875 

20.8566536 

7.5769849 

.002298851 

436 

190096 

82881856 

20.8806130 

7.5827865 

.002293578 

437 

190969 

83453453 

20.9045450 

7.5885793 

.002288330 

438 

191844 

84027672 

20.9284495 

7.5943633 

.002283105 

439 

192721 

84604519 

20.9523268 

7.6001385 

.002277904 

440 

193600 

85184000 

20.9761770 

7.6059049 

.002272727 

441 

194481 

85766121 

21.0000000 

7.6116626 

.002267574 

442 

195364 

86350888 

21.0237960 

7.6174116 

.002262443 

443 

196249 

86938307 

21.0475652 

7.6231519 

.002257336 

444 

197136 

8752S384 

21.0713075 

7.6288837 

.002252252 

445 

198025 

88121125 

21.0950231 

7.6346067 

.002247191 

446 

198916 

88716536 

21.1187121 

7.6403213 

.002242152 

447 

199S09 

89314623 

21.1423745 

7.6460272 

.002237136 

443 

200704 

89915392 

21.1660105 

7.6517247 

.002232143 

449 

201601 

90518849 

21.1896201 

7.6574138 

.002227171 

450 

202500 

91125000 

21.2132034 

7.6630943 

.002222222 

451 

203401 

91733851 

21.2367606 

7.6687665 

.002217295 

452 

204304 

92345408 

21.2602916 

7.6744303 

.002212389 

453 

205209 

92959677 

21.2837967 

7.6800857 

.002207506 

454 

206116 

93576664 

21.3072758 

7.6857328 

.002202643 

455 

207025 

94196375 

21.3307290 

7.6913717 

.002197802 

456 

207936 

94818816 

21.3541565 

7.6970023 

.002192982 

457 

208849 

95443993 

21.3775583 

7.7026246 

.002188184 

458 

209764 

96071912 

21.4009346 

7.7082388 

.002183406 

459 

210681 

96702579 

21.4242853 

7.7138448 

.002178649 

460 

211600 

97336000 

21.4476106 

7.7194426 

.002173913 

461 

212521 

97972181 

21.4709106 

7.7250325 

.002169197 

462 

213444 

98611128 

21.4941853 

7.7306141 

.002164502 

463 

214369 

99252847 

21.5174348 

7.7361877 

.002159827 

464 

215296 

99897344 

21.5406592 

7.7417532 

.002155172 

465 

216225 

100544625 

21.5638587 

7.7473109 

.002150538 

466 

217156 

101194696 

21.5870331 

7.7528606 

.002145923 

467 

218089 

101847563 

21.6101828 

7.7584023 

.002141328 

468 

219024 

102503232 

21.6333077 

7.7639361 

.002136752 

469 

219961 

103161709 

21.6564078 

7.7694620 

.002132196 

470 

220900 

103823000 

21.6794834 

7.7749801 

.002127660 

471 

221841 

104487111 

21.7025344 

7.7804904 

.002123142 

472 

222784 

105154048 

21.7255610 

7.7859928 

.002118644 

473 

223729 

105823817 

21.7485632 

7.7914875 

.002114165 

474 

224676 

106496424 

21.7715411 

7.7969745 

.002109705 

475 

225625 

107171875 

21.7944947 

7.8024538 

.002105263 

476 

226576 

107850176 

21.8174242 

7.8079254 

.002100840 

477 

227529 

108531333 

21.8403297 

7.8133392 

.002096436 

478 

228484 

109215352 

21.8632111 

7.8188456 

.002092050 

479 

229441 

109902239 

21.8860686 

7.8242942 

.002087683 

480 

230400 

110592000 

21.9089023 

7.8297353 

.002083333 

481 

231361 

111284641 

21.9317122 

7.8351688 

.002079002 

’ 482 

232324 

111980168 

21.9544984 

7.8405949 

.002074689 

483 

233289 

112678587 

21.9772610 

7.8460134 

.002070393 

4.84 

234256 

113379904 

22.0000000 

7.8514244 

.002066116 

485 

235225 

114084125 

22.0227155 

7.8568281 

.002061856 

4 85 

236196 

114791256 

22.0454077 

7.8622242 

.002057613 

487 

237169 

115501303 

22.06S0765 

7.8676130 

.002053388 

488 

238144 

116214272 

22.0907220 

7.8729944 

.002049180 

439 

239121 

116930169 

22.1133444 

7.8783684 

.002044990 

490 

240100 . 

117649000 

22.1359436 

7.8837352 

.002040816 

491 

241081 

118370771 

22.1585198 

7.8890946 

.002036660 

492 

242064 

119095488 

22.1810730 

7.8944468 

.002032520 

493 

. 243049 

119823157 

22.2036033 

7.8997917 

.002028398 

494 

244036 

120553784 

22.2261108 

7.9051294 

.002024291 

495 

245025 

121287375 

22.2485955 

7.9104599 

'.002020202 

496 

246016 

122023936 

22.2710575 

7.9157832 

.002016129 























146 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

~--' 

Reciprocals. 

497 

247009 

122763473 

22.2934968 

7.9210994 

.002012072 

493 

248004 

123505992 

22.3159136 

7.9264035 

.002008032 

499 

219001 

124251499 

22.3383079 

7.9317104 

.002004008 

500 

250000 

125000000 

22.3606793 

7.9370053 

.002000000 

501 

251001 

125751501 

22.3830293 

7.9422931 

.001996008 

502 

252004 

126506008 

22.4053565 

7.9475739 

.001992032 

503 

253009 

127263527 

22.4276615 

7.9528477 

.001938072 

504 

254016 

128024064 

22.4499443 

7.9581144 

.001984127 

505 

255025 

128787625 

22.4722051 

7.9633743 

.001930198 

506 

256036 

129554216 

22.4944438 

7.9686271 

.001976285 

507 

257049 

130323843 

22.5166605 

7.9733731 

.001972387 

503 

253064 

131096512 

22.5338553 

7.9791122 

.001968504 

509 

259031 

131872229 

22.5610283 

7.9843444 

.001964637 

510 

260100 

132651000 

22.5831796 

7.9895697 

.001960784 

511 

261121 

133432831 

22.6053091 

7.9947883 

.001956947 

512 

262144 

134217728 

22.6274170 

8.0000000 

.001953125 

513 

263169 

135005697 

22.6495033 

8.0052049 

.001949318 

514 

264196 

135796744 

22.6715631 

8.0104032 

.001945525 

515 

265225 

136590375 

22.6936114 

8.0155946 

.001941748 

516 

266256 

137388096 

22.7156334 

8.0207794 

.001937984 

517 

267239 

133188413 

22.7376340 

8.0259574 

.001934236 

518 

263324 

133991832 

22.7596134 

8.0311287 

.001930502 

519 

269361 

139798359 

22.7815715 

8.0362935 

.001926782 

520 

270400 

140608000 

22.8035085 

8.0414515 

.001923077 

521 

271441 

141420761 

22.8254244 

8.0466030 

.001919386 

522 

272484 

142236643 

22.8473193 

8.0517479 

.001915709 

523 

273529 

143055667 

22.8691933 

8.0568862 

.001912046 

524 

274576 

143377324 

22.8910463 

8.0620180 

.001903397 

525 

275625 

144703125 

22.9128785 

8.0671432 

.001904762 

526 

276676 

145531576 

22.9346899 

8.0722620 

.001901141 

527 

277729 

146363183 

22.9564806 

8.0773743 

.001897533 

523 

278734 

147197952 

22.9782506 

8.0824800 

.001893939 

529 

279341 

143035889 

23.0000000 

8.0875794 

.001890359 

530 

230900 

14S877000 

23.0217289 

8.0926723 

.001886792 

531 

231961 

149721291 

23.0434372 

8.0977589 

.001883239 

532 

233024 

150568763 

23.0651252 

8.1028390 

.001879699 

533 

284039 

151419437 

23.0867923 

8.1079128 

.001876173 

534 

235156 

152273304 

23.1034400 

8.1129803 

.001872659 

535 

236225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

237296 

153990656 

23.1516738 

8.1230962 

.001865672 

537 

238369 

154854153 

23.1732605 

8.1281447 

.001862197 

533 

289444 

155720872 

23.1948270 

8.1331870 

.001858736 

539 

290521 

156590819 

23.2163735 

8.1382230 

.001855288 

540 

291600 

157464000 

23.2379001 

8.1432529 

.001851852 

541 

292681 

158340421 

23.2594067 

8.1482765 

.001848429 

542 

293764 

159220033 

23.2808935 

S. 1532939 

.0018*15018 

543 

294849 

160103007 

23.3023604 

8.1583051 

.001841621 

544 

295936 

1609891S4 

23.3238076 

8.1633102 

.001838235 

545 

297025 

161878625 

23.3452351 

8.1683092 

.001834862 

546 

293116 

162771336 

23.3666429 

8.1733020 

.001831502 

547 

299209 

163667323 

23.3380311 

8.1782888 

.001828154 

548 

300304 

164566592 

23.4093998 

8.1832695 

.001824818 

549 

301401 

165469149 

23.4307490 

8.1882441 

.001821494 

550 

302500 

166375000 

23.4520788 

8.1932127 

.001818182 

551 

303601 

167234151 

23.4733392 

8.1931753 

.001814882 

552 

301704 

163196608 

23.4946802 

8.2031319 

.001811594 

553 

305809 

169112377 

23.5159520 

8.2080825 

.001808318 

554 

306916 

170031464 

23.5372046 

8.2130271 

.001805054 

555 

303025 

170953375 

23.5584380 

8.2179657 

.001801802 

556 

309136 

171879616 

23.5796522 

8.2228935 

.001798561 

557 

310249 

172308693 

23.6008474 

8.2278254 

.001795332 

558 

- 

311364 

173741112 

23.6220236 

8.2327463 

.001792115 






























CUBE ROOTS, AND RECIPROCALS 


14? 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

559 

312481 

174676879 

23.6431 SOS 

8.2376614 

.001783909 

560 

313600 

175616000 

23.6643191 

8.2425706 

.001785714 

561 

314721 

176558481 

23.6854386 

8.2474740 

.001782531 

562 

315844 

177504323 

23.7065392 

8.2523715 

.001779359 

563 

316969 

178453547 

23.7276210 

8.2572633 

.001776199 

561 

318096 

179406144 

23.7486842 

8.2621492 

.001773050 

565 

319225 

180362125 

23.7697286 

8.2670*294 

.001769912 

566 

320356 

181321496 

23.7907545 

8.2719039 

.001766784 

567 

321489 

182284263 

23.8117618 

8.2767726 

.001763668 

56S 

322624 

183250432 

23.8327506 

8.2816355 

.001760563 

569 

323761 

1842200JJ 

23.8537209 

8.2664928 

.001757469 

570 

324900 

185193000 

23.8746728 

8.2913444 

.001754386 

571 

326011 

186169411 

23.8956063 

8.2961903 

.001751313 

572 

327184 

187149248 

23.9165215 

8.3010304 

.01)1748252 

573 

326329 

188132517 

23.9374184 

8.3058651 

.001745201 

574 

329176 

189119224 

23.9582971 

8.3106941 

.001742160 

575 

330625 

190109375 

23.9791576 

8.3155175 

.001739130 

576 

331776 

191102976 

24.0000000 

8.3203353 

.001736111 

577 

332929 

192100033 

24.0208243 

8.3251475 

.001733102 

578 

334084 

193100552 

24.0416306 

8.3299542 

.001730104 

579 

335241 

194104539 

24.0624188 

S. 3347553 

.001727116 

580 

336400 

195112000 

24.0831891 

8.3395509 

.001724138 

581 

337561 

1.96122941 

24.1039416 

8.3443410 

.001721170 

582 

338724 

197137368 

24.1246762 

8.3191256 

.001718213 

583 

339889 

198155287 

24.1453929 

8.3539047 

.001715266 

5S4 

341056 

199176704 

24.1660919 

8.3586784 

.001712329 

585 

34222$ 

200201625 

24.1867732 

8.3634466 

.001709402 

586 

343396 

201230056 

24.2074369 

8.3632095 

.0017064S5 

587 

344569 

202262003 

24.2280829 

8.3729668 

.001703578 

588 

345744 

203297472 

24.2437113 

8.3777188 

.001700680 

589 

346921 

204336469 

24.2693222 

S.3824653 

.001697793 

590 

348100 

205379000 

24.2899156 

8.3872065 

.001694915 

591 

349231 

206425071 

21.3104916 

8.3919423 

.001692047 

592 

350464 

207474688 

24.3310501 

8.3966729 

.001689189 

593 

351649 

208527857 

24.3515913 

8.4013981 

.001636341 

594 

352836 

209584584 

24.3721152 

8.4061180 

.001683502 

595 

354025 

210644875 

24.3926218 

8.4108326 

.001680672 

596 

355216 

211708736 

24.4131112 

8.4155419 

.001677852 

597 

356409 

212776173 

24.4335834 

8.4202460 

.001675042 

598 

357604 

213847192 

24.4540385 

8.4249448 

.001672241 

599 

358301 

214921799 

24.4744765 

8.4296383 

.001669449 

600 

360000 

216000000 

24.4948974 

8.4343267 

.001666667 

601 

361201 

217081801 

24.5153013 

8.4390098 

.001663894 

602 

362404 

218167208 

24.5356883 

8.4436877 

.001661130 

603 

363609 

219256227 

24.5560583 

8.4483605 

.001658375 

604 

361816 

220348864 

24.5764115 

8.4530281 

.001655629 

605 

366025 

221445125 

24.5967478 

8.4576906 

.001652893 

606 

367236 

222545016 

24.6170673 

8.4623479 

.001650165 

607 

363449 

223643543 

24.6373700 

8.4670001 

.001647446 

608 

369664 

224755712 

24.6576560 

8.4716471 

.001644737 

609 

370881 

225866529 

24.6779254 

8.4762892 

.001642036 

610 

372100 

226981000 

24.6981781 

8.4809261 

.001639344 

611 

373321 

228099131 

24.7184142 

8.4855579 

.001636661 

612 

374544 

229220928 

24.7386338 

8.4901848 

.001633987 

613 

375769 

230346397 

24.7588368 

8.4948065 

.001631321 

614 

376996 

231475544 

24.7790234 

8.4994233 

.001628664 

615 

378225 

232603375 

24.7991935 

8.5040350 

.001626016 

616 

379456 

233744896 

24.8193473 

8.5086417 

.001623377 

617 

330639 

234885113 

24.8394847 

8.5132435 

.001620746 

618 

331924 

236029032 

24.8596058 

8.5178403 

.001618123 

619 

333161 

237176659 

24.8797106 

8.5224321 

.001615509 

620 

384400 

238328000 

24.8997992 

8.5270189 

.001612303 
























148 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

621 

335611 

239433061 

24.9198716 

8.5316009 

.001610306 

622 

336334 

240641343 

24.9399278 

8.5361780 

.001607717 

623 

333129 

241804367 

24.9599679 

8.5407501 

.001605136 

624 

339376 

242970624 

24.9799920 

8.5453173 

.001602564 

625 

39)625 

214140625 

25.0000000 

8.5493797 

.001600000 

626 

391376 

245314376 

25.0199920 

8.5544372 

.001597444 

627 

393129 

246491883 

25.0399631 

8.5589899 

.001594896 

628 

394334 

247673152 

25.0599282 

8.5635377 

.001592357 

629 

395641 

248858189 

25.0793724 

8.5680807 

.001589325 

630 

396900 

250047000 

25.0998003 

8.5726189 

.001537302 

631 

393161 

251239591 

25.1197134 

8.5771523 

.001584786 

632 

399424 

252435968 

25.1396102 

8.5816309 

.001582278 

633 

4006S9 

253636137 

25.1594913 

8.5862047 

.001579779 

634 

401956 

254840104 

25.1793566 

8.5907238 

.001577287 

635 

403225 

256047875 

25.1992063 

8.5952330 

.001574803 

636 

404496 

257259456 

25.2190404 

8.5997476 

.001572327 

637 

405769 

258474853 

25.2338539 

8.6042525 

.001569859 

633 

407044 

259894072 

25.2586619 

8.6037526 

.001567393 

639 

403321 

280917119 

25.2784493 

8.6132480 

.001564945 

640 

409600 

262144000 

25.2932213 

8.6177383 

.001562500 

641 

410381 

263374721 

25.3179778 

8.6222248 

.001560062 

612 

412164 

264609238 

25.3377189 

8.6267063 

.001557632 

613 

413449 

265347707 

25.3574447 

8.6311830 

.001555210 

614 

414736 

267089934 

25.3771551 

8.6356551 

.001552795 

615 

416025 

263336125 

25.3963502 

8.6401226 

.001550388 

616 

417316 

269586136 

25.4165301 

8.6445855 

.001547988 

647 

418609 

270340023 

25.4361947 

8.6490437 

.001545595 

643 

419904 

272097792 

25.4558441 

8.6534974 

.001543210 

649 

421201 

273359449 

25.4754784 

8.6579465 

.001540832 

650 

422500 

274625000 

25.4950976 

8.6623911 

.001538462 

651 

423301 

275394451 

25.5147016 

8.6663310 

.001536098 

652 

425101 

277167303 

25.5342907 

8.6712665 

.001533742 

653 

426109 

278445077 

25.5538647 

8.6756974 

.001531394 

654 

427716 

279726264 

25.5734237 

8.6301237 

.001529052 

655 

429025 

231011375 

25.5929678 

8.6845456 

.001526718 

656 

430336 

232300416 

25.6124969 

8.6389630 

.001524390 

657 

431649 

233593393 

25.6320112 

8.6933759 

.001522070 

653 

432961 

234890312 

25.6515107 

8.6977843 

.001519757 

659 

431231 

236191179 

25.6709953 

8.7021882 

.001517451 

660 

435600 

2S7496000 

25.6904652 

8.7065377 

.001515152 

681 

436921 

288804781 

25.7099203 

8.7109827 

.001512359 

662 

433244 

290117528 

25.7293607 

8.7153734 

.001510574 

663 

439569 

291434247 

25.7487864 

8.7197596 

.001508296 

664 

440396 

292754944 

25.7631975 

8.7241414 

.001506024 

665 

442225 

294079625 

25.7875939 

8.7235187 

.001503759 

666 

443556 

295403296 

25.8069758 

8.7328918 

.001501502 

667 

444399 

296740963 

25.8263431 

8.7372604 

.001499250 

663 

446224 

293077632 

25.8456960 

8.7416246 

.001497006 

669 

447561 

299418309 

25.8650343 

8.7459S46 

.001494763 

670 

443900 

300763000 

25.8343532 

8.7503401 

.001492537 

671 

450241 

302111711 

25.9036677 

8.7546913 

.001490313 

672 

451534 

303464443 

25.9229628 

8.7590383 

.001488095 

673 

452929 

304821217 

25.9422435 

8.7633809 

.0014S5S84 

674 

451276 

306182024 

25.9615100 

8.7677192 

.001433630 

675 

455625 

307546375 

25.9807621 

8.7720532 

.001481481 

676 

456976 

303915776 

26.0000000 

8.7763830 

.001479290 

677 

453329 

310283733 

26.0192237 

8.7807084 

.001477105 

678 

459684 

311665752 

26.0384331 

8.7850296 

.001474926 

679 

461041 

313046339 

26.0576284 

8.7893466 

.001472754 

630 

462400 

314432000 

26.0763096 

8.7936593 

.0014705SS 

631 

463761 

315821241 

26.0959767 

8.7979679 

.001468429 

632 

465121 

317214563 

26.1151297 

8.8022721 

.001466276 




















CUBE ROOTS, AND RECIPROCALS 


149 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

633 

466489 

31S6119S7 

26.1342687 

8.8065722 

.001464129 

634 

467856 

320013504 

26.1533937 

8.8108681 

.001461988 

685 

469225 

321419125 

26.1725047 

8.8151598 

.001459854 

686 

470596 

322S28856 

26.1916017 

8.8194474 

.001457726 

687 

471969 

324242703 

26.2106848 

8.8237307 

.001455604 

633 

473344 

325660672 

26.2297541 

8.8280099 

.001453488 

639 

474721 

327082769 

26.2488095 

8.8322850 

.001451379 

690 

476100 

328509000 

26.2678511 

8.8365559 

.001449275 

691 

477481 

329939371 

26.2868789 

8.8408227 

.001447178 

692 

478864 

331373888 

26.3058929 

8.8450854 

.001445087 

693 

480249 

332812557 

26.3248932 

8.8493440 

.001443001 

694 

481636 

334255384 

26.3438797 

3.8535985 

.001440922 

695 

483025 

335702375 

26.3628527 

8.8578489 

.001433849 

696 

484416 

337153536 

26.3818119 

8.8620952 

.001436782 

697 

485809 

33S608S73 

26.4007576 

8.8663375 

.001434720 

698 

487204 

340068392 

26.4196896 

8.8705757 

.001432665 

699 

488601 

341532099 

26.4386081 

8.8748099 

.001430615 

700 

490000 

343000000 

26.4575131 

8.8790400 

.001428571 

701 

491401 

344472101 

26.4764046 

8.8832661 

.001426534 

702 

492804 

345948408 

26.4952826 

8.8874882 

.001424501 

703 

494209 

347428927 

26.5141472 

8.8917063 

.001422475 

704 

495616 

348913664 

26.5329983 

8.8959204 

.001420455 

705 

497025 

350402625 

26.5518361 

8.9001304 

.001418440 

706 

498436 

351S95816 

26.5706605 

8.9043366 

.001416431 

707 

499849 

353393243 

26.5894716 

8.9085387 

.001414427 

70S 

501264 

354894912 

26.6082694 

8.9127369 

.001412429 

709 

502681 

356400829 

26.6270539 

8.9169311 

.001410437 

710 

504100 

357911000 

26.6458252 

8.9211214 

.001408451 

711 

505521 

359425431 

26.6645833 

8.9253078 

.001406470 

712 

506944 

360944128 

26.6833281 

8.9294902 

.001404494 

713 

508369 

362467097 

26.7020593 

8.9336687 

.001402525 

714 

509796 

363994344 

26.7207784 

8.9378433 

.001400560 

715 

511225 

365525875 

26.7394839 

8.9420140 

.001398601 

716 

512656 

367061696 

26.7581763 

8.9461809 

.001396648 

717 

514089 

363601813 

26.7768557 

8.9503438 

.001294700 

718 

515524 

370146232 

26.7955220 

8.9545029 

001392758 

719 

516961 

371694959 

26.8141754 

8.9586581 

.001390821 

720 

518400 

373248000 

26.8328157 

8.9628095 

.001388889 

721 

519841 

374805361 

26.S514432 

8.9669570 

.001386963 

722 

521284 

376367048 

26.8700577 

8.9711007 

.001385042 

723 

522729 

377933067 

26.8886593 

8.9752406 

.001383126 

724 

524176 

379503424 

26.9072481 

8.9793766 

.001381215 

725 

525625 

331078125 

26.9258240 

8.9835089 

.001379310 

726 

527076 

382657176 

26.9143872 

8.9876373 

.001377410 

727 

•528529 

334240583 

26.9629375 

8.9917620 

.001375516 

728 

529984 

335828352 

26.9814751 

8.9958829 

.001373626 

729 

531441 

337420439 

27.0000000 

9.0000000 

.001371742 

730 

532900 

389017000 

27.0185122 

9.0041134 

.001369863 

731 

534361 

390617891 

27.0370117 

9.0082229 

.001367989 

732 

535824 

392223168 

27.0554985 

9.0123288 

.001366120 

733 

537239 

393832837 

27.0739727 

9.0164309 

.001364256 

734 

538756 

395446904 

27.0924344 

9.0205293 

.001362398 

735 

540225 

397065375 

27.1103834 

9.0246239 

.001360544 

736 

541696 

398688256 

27.1293199 

9.0287149 

.001358696 

737 

543169 

400315553 

27.1477439 

9.0328021 

.001356852 

738 

544644 

401947272 

27. i 661554 

9.0368857 

.001355014 

739 

546121 

403583419 

27.1845544 

9.0409655 

.0013531S0 

740 

547600 

405224000 

27.2029410 

9.0450419 

.001351351 

741 

549081 

406869021 

27.2213152 

9.0491142 

.001349528 

742 

550564 

408518488 

27.2396769 

9.0531831 

.001347709 

743 

552019 

410172407 

27.2580263 

9.0572482 

.001345895 

714 

553536 

411830784 

27.2763634 

9.0613098 

.001344086 























150 TABLE XI SQUARES, CUBES, SQUARE ROOTS, 


j No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

T, • i 1 

Reciprocals. | 

745 

555025 

413493625 

27.2946381 

9.0653677 

.001342282 

746 

556516 

415160936 

27.3130006 

9.0694220 

.001340483 

747 

558009 

416832723 

27.3313007 

9.0734726 

.001333638 

743 

559501 

418503992 

27.3495887 

9.0775197 

.001336893 

749 

561001 

4201S9749 

27.3678644 

9.0815631 

.001335113 

750 

562500 

421375000 

27.3861279 

9.0856030 

.001333333 

751 

564001 

423564751 

27.4043792 

9.0896392 

.001331558 

752 

565504 

425259008 

27.4226184 

9.0936719 

.001329737 

753 

567009 

426957777 

27.4403455 

9.0977010 

.001328021 

754 

563516 

42S66I064 

27.4590604 

9.1017265 

.001326260 

755 

570025 

430363875 

27.4772633 

9.1057485 

.001324503 

756 

571536 

432081216 

27.4954542 

9.1097669 

.001322751 

757 

573049 

433798093 

27.5136330 

9.1137818 

.001321004 

753 

574564 

435519512 

27.5317993 

9.1177931 

.001319261 

759 

576031 

437245479 

27.5499546 

9.1218010 

.001317523 

760 

577600 

433976000 

27.5630975 

9.1258053 

.001315789 

761 

579121 

440711031 

27.5862284 

9.1293061 

.001314060 

762 

530614 

442450723 

27.6)43475 

9.1333034 

.001312336 

763 

582169 

444191947 

27.6224546 

9.1377971 

.001310616 

761 

583696 

445943744 

27.6405499 

9.1417874 

.001303901 

765 

585225 

447697125 

27.65S6334 

9.1457742 

.001307190 

766 

586756 

449455096 

27.6767050 

9.1497576 

.001305433 

767 

583239 

451217663 

27.6947648 

9.1537375 

.001303781 

763 

589324 

452934832 

27.7123129 

9.1577139 

.001302083 

769 

591361 

454756609 

27.7308492 

9.1616369 

.001300390 

770 

592900 

456533000 

27.7483739 

9.1656565 

.001298701 

771 

594441 

453314011 

27.7653863 

9.1696225 

.001297017 

772 

595984 

460099648 

27.7848880 

9.1735852 

.001295337 

773 

597529 

461839917 

27.802S775 

9.1775445 

.001293661 

774 

599076 

463694824 

27.8203555 

9.1815003 

.001291990 

775 

600625 

465484375 

27.8388218 

9.1854527 

.001290323 

776 

602176 

467288576 

27.8567766 

9.1894018 

.00128S660 

777 

603729 

469097433 

27.8747197 

9.1933474 

.001287001 

773 

605234 

470910952 

27.8926514 

9.1972397 

.0012S5347 

779 

606341 

472729139 

27.9105715 

9.2012286 

.001283697 

780 

603400 

474552000 

27.9234801 

9.2051641 

.001282051 

781 

609961 

476379541 

27.9463772 

9.2090962 

.001280410 

732 

611524 

473211763 

27.3542629 

9.2130250 

.001278772 

783 

613039 

480043637 

27.9321372 

9.2169505 

.001277139 

784 

614656 

481890301 

28.0000000 

9.2203726 

.001275510 

735 

616225 

483736625 

23.0178515 

9.2247914 

.001273885 

786 

617796 

4355S7656 

23.0356915 

9.2287063 

.001272265 

737 

619369 

487443403 

28.0535203 

9.2326189 

.001270648 

733 

620944 

439303372 

28.0713377 

9.2365277 

.001269036 

789 

622521 

491169069 

23.0891433 

9.2404333 

.001267427 

790 

624100 

493039000 

23.1069.336 

9.2443355 

.001265823 

791 

625631 

494913671 

28.1247222 

9.2482344 

.001264223 

792 

627264 

496793088 

23.1424946 

9.2521300 

.001262626 

793 

628349 

493677257 

28.1602557 

9.2560224 

.001261034 

791 

639136 

500566184 

23.1780056 

9.2599114 

.001259446 

795 

632025 

502459375 

23.1957444 

9.2637973 

.001257862 

796 

633616 

504358336 

28.2134720 

9.2676793 

.001256281 

797 

635209 

506261573 

28.2311884 

9.2715592 

.001254705 

793 

636304 

508169592 

28.248S933 

9.2754352 

.001253133 

799 

633401 

510032399 

28.2665831 

9.2793081 

.001251564 

800 

640000 

512000000 

28.2342712 

9.2331777 

.001250000 ' 

801 

641601 

513922401 

28.30194:34 

9.2870440 

.001248439 

802 

643204 

515849608 

28.3196045 

9.2909072 

.001246383 

803 

644309 

517781627 

23.3372546 

9.2947671 

.001245330 

804 

646416 

519713464 

23.354S933 

9.2936239 

.001243781 

805 

648025 

521680125 

23.3725219 

9.3024775 

.001242236 

806 

649636 

523606616 

23.3901391 

9.3063273 

.001240695 





























151 


CUBE ROOTS, AND RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

807 

651249 

525557943 

23.4077454 

9.3101750 

.001239157 

803 

652364 

527514112 

28.4253408 

9.3140190 

.001237624 

800 

654481 

529475129 

28.4429253 

9.3178599 

.001236094 

810 

656100 

531441000 

28.4601989 

9.3216975 

.001234563 

811 

657721 

533411731 

23.4780617 

9.3255320 

.001233046 

812 

659344 

535337328 

23.4956137 

9.3293634 

.001231527 

813 

650969 

537367797 

28.5131549 

9.333J9I6 

.001230012 

814 

662596 

539353144 

28.5306352 

9.3370167 

.001228501 

815 

664225 

541343375 

28.5432048 

9.3408386 

.001226994 

816 

665856 

543338496 

23.5657137 

9.3446575 

.001225490 

817 

6674S9 

545333513 

23.5832119 

9.3484731 

.001223990 

818 

669124 

547343432 

28.6006993 

9.3522357 

.001222494 

81S 

670761 

549353259 

28 6181760 

9.3560952 

.001221001 

820 

672400 

551363000 

29.6356421 

9.3599016 

.001219512 

821 

674041 

553337661 

23.6530976 

9.3637049 

.001218027 

822 

675634 

555412248 

23.6705424 

9.3675051 

.001216545 

823 

677329 

557441767 

23.6379766 

9.3713022 

.001215067 

824 

678976 

559476224 

23.7054002 

9.3750963 

.001213592 

825 

630625 

561515625 

28.7223132 

9.3788373 

.001212121 

826 

632276 

563559976 

23.7402157 

9.3326752 

.001210654 

827 

633929 

565609233 

28.7576077 

9.3364600 

.001209190 

823 

685534 

567663552 

23.7749391 

9.3902419 

.001207729 

829 

687241 

569722739 

28.7923601 

9.3940206 

.001206273 

830 

633900 

571787000 

23.8097206 

9.3977964 

.001204819 

831 

690561 

573856191 

23.8270708 

9.4015691 

.001203369 

832 

692224 

575930363 

23.8444102 

9.4053337 

.001201923 

833 

693339 

578009537 

28.8617394 

9.4091054 

.001200480 

834 

695556 

580093704 

28.8790582 

9.4128690 

.001199041 

835 

697225 

532182375 

23.8963666 

9.4166297 

.001197605 

836 

693396 

534277056 

23.9136646 

9.4203373 

.001196172 

837 

700569 

536376253 

28.9309523 

9.4241420 

.001194743 

833 

702244 

583430472 

23.9482297 

9.4278936 

.001193317 

839 

703921 

590589719 

28.9654967 

9.4316423 

.001191895 

840 

705600 

592704000 

28.9327535 

9.4353880 

.001190476 

841 

707231 

594823321 

29.0000000 

9.4391307 

.001189061 

842 

703964 

596947638 

29.0172363 

9.4428704 

.001187643 

843 

710649 

599077107 

29.0344623 

9.4466072 

.001186240 

844 

712336 

601211584 

29.0516731 

9.4503410 

.001184834 

845 

714025 

603351125 

29.0638837 

9.4540719 

.001183432 

846 

715716 

605495736 

29.0860791 

9.4577999' 

.001182033 

847 

717409 

607645423 

29.1032644 

9.4615249 

.001180638 

843 

719104 

609800192 

29.1204396 

9.4652470 

.001179245 

849 

720301 

611960049 

29.1376046 

9.4639661 

.001177856 

850 

722500 

614125000 

29.1547595 

9.4726824 

.001176471 

851 

724201 

616295051 

29.1719043 

9.4763957 

.001175088 

852 

725904 

618470203 

29.1S90390 

9.4801061 

.001173709 

853 

727609 

620650477 

29.2061637 

9.4833135 

.001172333 

854 

729316 

622335364 

29.2232784 

9.4875182 

.001170960 

855 

731025 

625026375 

29.2403330 

9.4912200 

.001169591 

856 

732736 

627222016 

29.2574777 

9.4949188 

.001168224 

857 

734449 

629122793 

29.2745623 

9.4986147 

.001166861 

858 

736164 

631628712 

29.2916370 

9.5023078 

.001165501 

859 

737831 

633339779 

29.3037013 

9.5059930 

.001164144 

860 

739600 

636056000 

29.3257566 

9.5096854 

.001162791 

861 

_ 741321 

633277331 

29.3423015 

9.5133699 

.001161440 

862 

743044 

640503923 

29.3598365 

9.5170515 

.001160093 

863 

744769 

642735647 

29.3763616 

9.5207303 

.001158749 

864 

746496 

644972544 

29.3933769 

9.5244063 

.001157407 

865 

743225 

647214625 

29.4103823 

9.5230794 

.001156069 

866 

749956 

619461896 

29.4278779 

9.5317497 

.001154734 

867 

751639 

651714363 

29.4448637 

9.5354172 

.001153403 

863 

753424 

653972032 

29.4618397 

9.5390S13 

.001152074 























152 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals 

869 

755161 

656234909 

29.4788059 

9.5427437 

.001150748 

870 

756900 

658503000 

29.4957624 

9.5464027 

.001149425 

871 

758641 

660776311 

29.5127091 

9.5500589 

.001148106 

872 

760384 

663054848 

29.5296461 

9.5537123 

.001146789 

873 

762129 

665338617 

29.5465734 

9.5573630 

.003145475 

874 

763876 

667627624 

29.5634910 

9.5610108 

.001144165 

875 

765625 

669921875 

29.5803989 

9.5646559 

.001142857 

876 

767376 

672221376 

29.5972972 

9.5682982 

.001141553 

877 

769129 

674526133 

29.6141858 

9.5719377 

.001140251 

878 

770884 

676836152 

29.6310648 

9.5755745 

.00113S952 

879 

772641 

679151439 

29.6479342 

9.5792085 

.001137656 

880 

774400 

681472000 

29.6647939 

9.5828397 

.001136364 

881 

776161 

683797841 

29.6816442 

9.5864682 

.001135074 

882 

777924 

686128968 

29.6984848 

9.5900939 

.001133787 

883 

7796S9 

688465337 

29.7153159 

9.5937169 

.001132503 

884 

781456 

690807104 

29.7321375 

9.5973373 

.001131222 

S85 

783225 

693154125 

29.7489496 

9.6009548 

.001129944 

886 

784996 

695506456 

29.7657521 

9.6045696 

.001128668 

887 

786769 

697864103 

29.7825452 

9.6081817 

.001127396 

838 

788544 

700227072 

29.7993239 

9.6117911 

.001126126 

889 

790321 

702595369 

29.8161030 

9.6153977 

.001124859 

890 

792100 

704969000 

29.8328678 

9.6190017 

.001123596 

891 

793881 

707347971 

29.8496231 

9.6226030 

.001122334 

892 

795664 

709732288 

29.8663690 

9.6262016 

.001121076 

893 

797449 

712121957 

29.8831056 

9.6297975 

.001119821 

894 

799236 

714516984 

29.8998328 

9.6333907 

.001118568 

895 

801025 

716917375 

29.9165506 

9.6369812 

.001117318 

896 

802S16 

719323136 

29.9332591 

9.6405690 

.001116071 

897 

804609 

721734273 

29.9499583 

9.6441542 

.001114827 

898 

806404' 

724150792 

29.9666481 

9.6477367 

.0011135S6 

899 

808201 

726572699 

29.9833287 

9.6513166 

.001112347 

900 

810000 

729000000 

30.0000000 

9.6548938 

.001111111 

901 - 

811801 

731432701 

30.0166620 

9.6584684 

.001109878 

902 

813604 

733870808 

30.0333148 

9.6620403 

.001108647 

903 

815409 

736314327 

30.0499584 

9.6656096 

.001107420 

904 

817216 

738763264 

30.0665928 

9.6691762 

.001106195 

905 

819025 

741217625 

30.0832179 

9.6727403 

.001104972 

906 

820836 

743677416 

30.0998339 

9.6763017 

.001103753 

907 

822649 

746142643 

30.1164407 

9.6798604 

.001102536 

908 

824464 

748613312 

30.1330383 

9.6S34166 

.001101322 

909 

826281 

751089429 

30.1496269 

9.6869701 

.001100110 

910 

828100 

753571000 

30.1662063 

9.6905211 

.001098901 

911 

829921 

756058031 

30.1827765 

9.6940694 

.001097695 

912 

831744 

758550528 

30.1993377 

9.6976151 

.001096491 

913 

833569 

761048497 

30.2158899 

9.7011583 

.001095290 

914 

835396 

763551944 

30.2324329 

9.7046989 

.001094092 

915 

837225 

766060875 

30.2489669 

9.7082369 

.001092896 

916 

839056 

768575296 

30.2654919 

9.7117723 

.001091703 

917 

840389 

771095213 

30.2820079 

9.7153051 

.001090513 

918 

842724 

773620632 

30.2985148 

9.7188354 

.001089325 

919 

844561 

776151559 

30.3150128 

9.7223631 

.001088139 

920 

846400 

778688000 

30.3315018 

9.7258883 

.001086957 

921 

848241 

781229961 

30.3479818 

9.7294109 

.001085776 

922 

850084 

783777443 

30.3644529 

9.7329309 

.001084599 

923 

851929 

786330467 

30.3809151 

9.7364484 

.001083423 

924 

853776 

788889024 

30.3973633 

9.7399634 

.001082251 

925 

855625 

791453125 

30.4133127 

9.7434758 

.0010810S1 

926 

857476 

794022776 

30.4302481 

9.7469857 

.001079914 

927 

859329 

796597S83 

30.4466747 

9.7504930 

.001078749 

928 

861184 

799178752 

30.4630924 

9.7539979 

.001077586 

929 

863041 

801765089 

30.4795013 

9.7575002 

.001076426 

930 

864900 

804357000 

30.4959014 

9.7610001 

.001075269 


zrl) 


































CUBE ROOTS, AND RECIPROCALS 


153 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

931 

866761 

806954491 

30.5122926 

9.7644974 

.001074114 

932 

863624 

809557568 

30.5286750 

9.7679922 

.001072961 

933 

870439 

812166237 

30.5450487 

9.7714845 

.001071811 

934 

872356 

814780504 

30.5614136 

9.7749743 

.001070664 

935 

874225 

817400375 

30.5777697 

9.7784616 

.001069519 

936 

876096 

820025356 

30.5941171 

9.7829466 

.001068376 

937 

877969 

822656953 

30.6104557 

9.7854288 

.001067236 

933 

879844 

825293672 

30.6267857 

9.7889087 

.001066098 

939 

881721 

827936019 

30.6431069 

9.7923861 

.001064963 

940 

833600 

830584000 

30.6594194 

9.7958611 

.001063830 

941 

835481 

833237621 

30.6757233 

9.7993336 

.001062699 

942 

837364 

835396888 

30.6920185 

9.8028036 

.001061571 

943 

889249 

838561807 

30.7083051 

9.8062711 

.001060445 

944 

891136 

841232384 

30.7245830 

9.8097362 

.001059322 

945 

893025 

843903625 

30.7408523 

9.8131989 

.001058201 

946 

894916 

846590536 

30.7571130 

9.8166591 

.001057082 

947 

896309 

849278123 

30.7733651 

9.8201169 

.001055966 

943 

898704 

851971392 

30.7896086 

9.8235723 

.001054852 

949 

900601 

854670349 

30.8058436 

9.8270252 

.001053741 

950 

902500 

857375000 

30.8220700 

9.8304757 

.001052632 

951 

904401 

860085351 

30.8382879 

9.8339238 

.001051525 

952 

906304 

862801408 

30.8544972 

9.8373695 

.001050420 

953 

903209 

865523177 

30.8706931 

9.8408127 

.001049318 

954 

910116 

863250664 

30.8868904 

9.8442536 

.001048218 

955 

912025 

870983875 

30.9030743 

9.8476920 

.001047120 

956 

913936 

873722816 

30.9192497 

9.8511280 

.001046025 

957 

915849 

876467493 

30.9354166 

9.8545617 

.001044932 

958 

917764 

879217912 

30.9515751 

9.8579929 

.001043841 

959 

919631 

831974079 

30.9677251 

9.8614218 

.001042753 

960 

921600 

884736000 

30.9838668 

9.8648483 

.001041667 

961 

923521 

887503631 

31.0000000 

9.8682724 

.001040533 

962 

925444 

890277128 

31.0161248 

9.8716941 

.001039501 

963 

927369 

893056347 

31.0322413 

9.8751135 

.001038422 

964 

929296 

895841344 

31.0483494 

9.8785305 

.001037344 

965 

931225 

898632125 

31.0644491 

9.8819451 

.001036269 

966 

933156 

901428696 

31.0805405 

9.8853574 

.001035197 

967 

935089 

904231063 

31.0966236 

9.8887673 

.001034126 

963 

937024 

907039232 

31.1126984 

9.8921749 

.001033058 

969 

933961 

909853209 

31.1287648 

9.8955801 

.001031992 

970 

940900 

912673000 

31.1448230 

9.8989830 

.001030928 

971 

942841 

915498611 

31.1608729 

9.9023835 

.001029866 

972 

944784 

918330048 

31.1769145 

9.9057817 

.001028307 

973 

946729 

921167317 

31.1929479 

9.9091776 

.001027749 

974 

948676 

924010424 

31.2089731 

9.9125712 

.001026694 

975 

950625 

926859375 

31.2249900 

9.9159624 

.001025641 

976 

952576 

929714176 

31.2409987 

9.9193513 

.001024590 

977 

954529 

932574833 

31.2569992 

9.9227379 

.001023541 

978 

956484 

935441352 

31.2729915 

9.9261222 

.001022495 

979 

958441 

938313739' 

31.2889757 

9.9295042 

.001021450 

9S0 

960400 

941192000 

31.3049517 

9.9328839 

.001020408 

981 

962361 

944076141 

31.3209195 

9.9362613 

.001019368 

982 

964324 

946966168 

31.3368792 

9.9396363 

.001018330 

983 

966239 

949862087 

31.3528308 

9.9430092 

.001017294 

984 

968256 

952763904 

31.3687743 

9.9463797 

.001016260 

985 

970225 

955671625 

31.3847097 

9.9497479 

.001015228 

986 

972196 

958585256 

31.4006369 

9.9531138 

.001014199 

98 7 

974169 

961504803 

31.4165561 

9.9564775 

.001013171 

938 

976144 

964430272 

31.4324673 

9.9598389 

.001012146 

939 

978121 

967361669 

31.4483704 

9.9631981 

.001011122 

990 

980100 

970299000 

31.4642654 

9.9665549 

.001010101 

991 

932081 

973242271 

31.4801525 

9.9699095 

.001009082 

992 

934064 

976191488 

31.4960315 

9.9732619 

.001008065 


8 




















154 


TABLE XI. SQUARES, CUBES, AC 


No. 

Squares. 

Cubes. 

Square Roots. 

Cube Roots. 

Reciprocals. 

993 

994 

995 

996 

997 

998 

999 

1000 

1001 

1002 

1003 

1004 

1005 

1006 

1007 

1008 

1009 

1010 
1011 
1012 

1013 

1014 

1015 

1016 

1017 

1018 

1019 

1020 
1021 
1022 

1023 

1024 

1025 

1026 

1027 

1028 

1029 

1030 

1031 

1032 

1033 

1034 

1035 

1036 

1037 

1038 

1039 

1040 

1041 

1042 

1043 

1044 

1045 

1046 

1047 

1048 

1049 

1050 

1051 

1052 

1053 

1054 

9S6049 

988036 

990025 

992016 

994009 

996004 

993001 

1000000 ■ 
1002001 
1004004 
1006009 
1008016 
1010025 
1012036 
1014049 
1016064 
1018081 

1020100 

1022121 

1024144 

1026169 

1028196 

1030225 

1032256 

1034289 

1036324 

1033361 

104040C 

1042441 

1044484 

1046529 

1048576 

1050625 

1052676 

1054729 

1056784 

1058841 

1060900 

1062961 

1065024 

1067089 

1069156 

1071225 

1073296 

1075369 

1077444 

1079521 

1081600 

1083681 

1085764 

1087849 

1089936 

1092025 

1094116 

1096209 

1098304 

1100401 

1102500 

1104601 

1106704 

1108809 

1110916 

979146657 

982107784 

985074875 

988047936 

991026973 

994011992 

997002999 

1000000000 

1003003001 

1006012008 

1009027027 

1012048064 

1015075125 

1018108216 

1021147343 

1024192512 

1027243729 

1030301000 

1033364331 

1036433728 

1039509197 

1042590744 

1045678375 

1048772096 

1051871913 

1054977832 

1058089859 

IU61208000 
1064332261 
1067462648 
1070599167 
1073741824 
1076890625 
1080045576 
1083206683 
1086373952 
1089547389 

1092727000 

1095912791 

1099104768 

1102302937 

1105507304 

1108717875 

1111934656 

1115157653 

1118386S72 

1121622319 

1124864000 

1128111921 

1131366088 

1134626507 

1137893184 
1141166125 
1144445336 
1147730823 
1151022592 
1154320649 

1157625000 

1160935651 

1164252608 

1167575877 

1170905464 

31.5119025 
31.5277655 
31.5436206 
31.5594677 
31.5753068 
31.5911380 
31.6069613 

31.6227766 

31.6385840 

31.6543836 

31.6701752 

31.6859590 

31.7017349 

31.7175030 

31.7332633 

31.7490157 

31.7647603 

31.7804972 

31.7962262 

31.8119474 

31.8276609 

31.8433666 

31.8590646 

31.8747549 

31.8904374 

31.9061123 

31.9217794 

31 9374388 
31.9530906 
31.9637347 
31.9843712 
32.0000000 
32.0156212 
32.0312348 
32.0468407 
32.0624391 
32.0780298 

32.0936131 

32.1091887 

32.1247568 

32.1403173 

32.1558704 

32.1714159 

32.1869539 

32.2024844 

32.2180074 

32.2335229 

32.2490310 

32.2645316 

32.2800248 

32.2955105 

32.3109888 

32.3264598 

32.3419233 

32.3573794 

32.3728281 

32.3882695 

32.4037035 

32.4191301 

32.4345495 

32.4499615 

32.4653662 

9.9766120 

9.9799599 

9.9833055 

9.9866488 

9.9899900 

9.9933289 

9.9966656 

10.0000000 

10.0033322 

10.0066622 

10.0099899 

10.0133155 

10.0166389 

10.0199601 

10.0232791 

10.0265958 

10.0299101 

10.0332228 

10.0365330 

10.0398410 

10.0431469 

10.0464506 

10.0497521 

10.0530514 

10.0563485 

10.0596435 

10.0629364 

10.0662271 

10.0695156 

10.0728020 

10.0760863 

10.0793684 

10.0826484 

10.0859262 

10.0892019 

10.0924755 

10.0957469 

10.0990163 

10.1022835 

10.1055487 

10.1088117 

10.1120726 

10.1153314 

10.1185882 

10.1218428 

10.1250953 

10.1283457 

10.1315941 

10.1348403 

10.1380845 

10.1413266 

10.1445667 

10.1478047 

10.1510406 

10.1542744 

10.1575062 

10.1607359 

10.1639636 

10.1671893 

10.1704129 

10.1736344 

10.1768539 

.001007049 

.001006036 

.001005025 

.001004016 

.001003009 

.001002004 

.001001001 

.001000000 

.0009990010 

.0009980040 

.0009970090 

.0009960159 

.0009950249 

.0009940358 

.0009930487 

.0009920635 

.0009910803 | 

.0009900990 i 

.0009891197 

.0009881423 

.0009871668 

.0009861933 

.0009852217 

.0009842520 

.0009S32842 

.0009823183 

.0009813543 

.0009803922 

.0009794319 

.0009784736 

.0009775171 

.0009765625 

.0009756098 

.0009746589 

.0009737098 

.0009727626 

.0009718173 

.0009708738 
.0009699321 
.0009689922 
.0009680542 
.0009671180 
.0009661836 
.0009652510 
.0009643202 
.0009633911 
.0009624639 

.0009615385 
.0009606148 
.0009596929 
.0009587738 
.0009578544 
.0009569378 
.0009560229 
.0009551098 
. 00095419S5 
.0009532888 

.0009523810 

.0009514748 

.0009505703 

.0009496676 

.0009487666 

























TABLE XII. 


LOGARITHMS OF NUMBERS 


FROM I TO 10,000 


156 


TABLE XII. LOGARITHMS OF NUMBERS 


No. 

0 

1 

2 

3 

4. 

5 

6 ) 

7 

8 

9 

DHf. 

100 

000000 

000434 

000868 

001301 

001734 

002166 

002598 

003029 

003461 

003891 

432 

I 

4321 

4751 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

428 

2 

8600 

9026 

9451 

9876 

010300 

010724 

011147 

011570 

011993 

012415 

424 

3 

012837 

013259 

0136S0 

014100 

4521 

4940 

5360 

5779 

6197 

6616 

420 

4 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

020361 

020775 

416 

5 

021189 

021603 

022016 

022428 

022841 

023252 

023664 

024075 

4486 

4896 

412 

6 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

408 

7 

9384 

9789 

030195 

030600 

031004 

031408 

031812 

032216 

032619 

033021 

404 

8 

033424 

033326 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

400 

9 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

040207 

040602 

040998 

397 

110 

041393 

041787 

042182 

042576 

042969 

043362 

043755 

044148 

044540 

044932 

393 

1 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

390 

2 

9218 

9606 

9993 

050380 

050766 

051153 

051538 

051924 

052309 

052694 

386 

3 

053078 

053463 

053846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

383 

4 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

060320 

379 

5 

060698 

061075 

061452 

061829 

062206 

062582 

062958 

063333 

063709 

4083 

376 

6 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

373 

7 

8186 

8557 

8928 

9298 

9668 

070038 

070407 

070776 

071145 

071514 

370 

8 

071882 

072250 

072617 

072985 

073352 

3718 

4085 

4451- 

4816 

5182 

366 

9 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

079181 

079543 

079904 

0S0266 

0S0626 

080987 

081347 

081707 

082067 

082426 

360 

1 

032785 

083144 

083503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

2 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

3 

9905 

090258 

090611 

090963 

091315 

091667 

092018 

092370 

092721 

093071 

352 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

5 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

100026 

346 

6 

100371 

100715 

101059 

101403 

101747 

102091 

102434 

102777 

103119 

3462 

343 

7 

3304 

4146 

4487 

4S2S 

5169 

5510 

5851 

6191 

6531 

6871 

341 

8 

7210 

7549 

78S8 

8227 

8565 

8903 

9241 

9579 

9916 

110253 

338 

9 

110590 

110926 

111263 

111599 

111934 

112270 

112605 

112940 

113275 

3609 

335 

130 

113943 

114277 

114611 

114944 

115278 

115611 

115943 

116276 

116608 

116940 

333 

1 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

95S6 

9915 

120245 

330 

2 

120574 

120903 

121231 

121560 

121888 

122216 

122544 

122871 

123198 

3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

4 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

130012 

323 

5 

130334 

130655 

130977 

131298 

131619 

131939 

132260 

132580 

132900 

3219 

321 

6 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

318 

7 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

316 

8 

9879 

140194 

140508 

140S22 

141136 

141450 

141763 

142076 

142389 

142702 

314 

9 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140 

146128 

146438 

146748 

147058 

147367 

147676 

147985 

14S294 

14S603 

148911 

309 

1 

9219 

9527 

9835 

150142 

150449 

150756 

151063 

151370 

151676 

1519S2 

307 

2 

152288 

152594 

152900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

3 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

4 

8362 

8664 

8965 

9266 

9567 

9868 

16016S 

160469 

160769 

161068 

301 

5 

161368 

161667 

161967 

162266 

162564 

162S63 

3161 

3460 

3758 

4055 

299 

6 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6130 

6726 

7022 

297 

7 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

93S0 

9674 

9968 

295 

8 

170262 

170555 

170848 

171141 

171434 

171726 

172019 

172311 

172603 

172895 

293 

9 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

150 

176091 

176381 

176670 

176959 

177248 

177536 

177825 

178113 

178401 

178689 

2S9 

1 

8977 

9264 

9552 

9839 

180126 

180413 

180699 

180986 

181272 

181558 

287 

2 

181844 

182129 

182415 

182700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

3 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

4 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

190051 

281 

5 

190332 

190612 

190S92 

191171 

191451 

191730 

192010 

192289 

192567 

2S46 

279 

6 

3125 

3403 

36S1 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

7 

5900 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

8 

8657 

8932 

9206 

9481 

9755 

200029 

200303 

200577 

200850 

201124 

274 

9 

201397 

201670 

201943 

202216 

202488 

2761 

3033 

3305 

3577 

3S48 

272 

No 

0 

1 

2 

3 

1 4 

5 

G 

7 

8 

9 

DHL t 























































TABLE XII. LOGARITHMS OF NUMBERS 


15 ? 


No. 

0 

1 

3 | 

3 

4t 

5 

6 1 

7 | 

8 

9 

Biff. 

160 

204120 

204391 

204663 

204934 

205204 

205475 

205746 

206016 

206236 

206556 

271 

1 

6325 

7096 

7365 

7634 

7904 

8173 

8441 

87101 

8979 

9247 

269 

2 

9515 

9783 

210051 

210319 

210536 

210353 

211121 

211338 

211654 

211921 

267 

3 

212183 

212154 

2720 

2936 

3252 

3518 

3783 

4049 

4314 

4579 

266 

4 

4344 

5109 

5373 

5633 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7434 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

6 

220103 

220370 

220631 

220892 

221153 

221414 

221675 

221936 

222196 

222456 

261 

7 

2716 

2976 

3235 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5563 

5326 

6034 

6342 

6600 

6358 

7115 

7372 

7630 

258 

a 

7387 

8144 

8400 

8657 

8913 

9170 

9426 

9632 

9938 

230193 

256 

170 

230449 

230704 

230960 

231215 

231470 

231724 

231979 

232234 

232488 

232742 

255 

i 

2996 

3250 

3501 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

' 2 

5523 

5781 

6033 

6235 

6537 

6789 

7041 

7292 

7544 

7795 

252 

3 

8046 

8297 

8543 

8799 

9049 

9299 

9550 

9800 

240050 

240300 

250 

4 

240549 

240799 

241048 

241297 

241546 

241795 

242044 

242293 

2541 

2790 

249 

5 

3033 

3286 

3531 

3732 

4030 

4277 

4525 

4772 

5019 

5266 

248 

6 

5513 

5759 

6006 

6252 

6199 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9637 

9932 

250176 

245 

8 

250420 

250664 

250903 

251151 

251395 

251633 

251881 

252125 

252368 

2610 

243 

9 

2353 

3096 

3333 

3530 

3322 

4064 

4306 

4548 

4790 

5031 

242 

180 255273 

255514 

255755 

255996 

256237 

256477 

256718 

256958 

257193 

257439 

241 

11 

•7679 

7918 

8153 

8393 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

250310 

260543 

260787 

261025 

261263 

261501 

261739 

261976 

262214 

233 

3 

2451 

2633 

2925 

3162 

3399 

3636 

3373 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

6 

9513 

9746 

9930 

270213 

270446 

270679 

270912 

271144 

271377 

271609 

233 

7 

271842 

272074 

272306 

2533 

2770 

3001 

3233 

3464 

3696 

3927 

232 

8 

4153 

4339 

4620 

4850 

5031 

5311 

5542 

5772 

6002 

6232 

230 

9 

6152 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278754 

278932 

27921l 

279439 

279667 

279395 

230123 

280351 

280578 

280806 

228 

1 

231033 

231261 

231488 

231715 

231942 

282169 

2396 

2622 

2849 

3075 

227 

2 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

3 

5557 

5782 

6007 

6232 

6456 

6631 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

5 

290035 

290257 

290480 

290702 

290925 

291147 

291369 

291591 

291813 

292034 

222 

6 

2256 

2478 

2699 

2920 

3141 

3363 

3534 

3304 

4025 

4246 

221 

7 

4466 

4637 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

8 

6665 

6384 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

9 

8353 

9071 

92S9 

9507 

9725 

9943 

300161 

300378 

300595 

300813 

218 

200 

301030 

301247 

301464 

301631 

301898 

302114 

302331 

302547 

302764 

302980 

217 

1 

3196 

3412 

3623 

3344 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5995 

6211 

6425 

6639 

6354 

7063 

7232 

215 

3 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

4 

9630 

9343 

310056 

310263 

310431 

310693 

310906 

311118 

311330 

311542 

212 

5 

311754 

311966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

6 

3367 

4078 

4239 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

7 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

8 

8083 

8272 

8431 

8639 

8393 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

320354 

320562 

320769 

320977 

321184 

321391 

321598 

321805 

322012 

207 

210 

322219 

322426 

322633 

322339 

323046 

323252 

323458 

323665 

323S71 

324077 

206 

1 

4232 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

8330 

8533 

8787 

8991 

9194 

9393 

9801 

9305 

330003 

330211 

203 

4 

330114 

330617 

330819 

331022 

331225 

331427 

331630 

331832 

2034 

2236 

202 

5 

2433 

2640 

2342 

3044 

3216 

3447 

3649 

3350 

4051 

4253 

202 

6 

4454 

4655 

4856 

5057 

5257 

5458 

5653 

5859 

6059 

6260 

201 

7 

6460 

6660 

6360 

7060 

7260 

7459 

7659 

7853 

8058 

8257 

200 

8 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9349 

340047 

340246 

199 

9 

340444 

310612 

340341 

341039 

341237 

341435 

341632 

341830 

2028 

2225 

198 

No 

0 

1 1 

3 

3 

4 - 

5 

6 

7 

8 

9 

Diff.j 





























































158 


TABLE XII. LOGARITHMS OF NUMBERS 


No. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Diff. 

220 

342423 

342620 

342817 

343014 

343212 

343409 

343606 

343802 

343999 

344196 

197 

1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

350054 

194 

4 

350248 

350442 

350636 

350829 

351023 

351216 

351410 

351603 

351796 

1989 

193 

5 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

6 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

8 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9260 

9456 

9646 

190 

9 

9835 

360025 

360215 

360404 

360593 

360783 

360972 

361161 

361350 

361539 

189 

230 

361728 

361917 

362105 

362294 

362482 

362671 

362S59 

363048 

363230 

363424 

188 

1 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

2 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

370143 

370328 

370513 

370698 

370883 

185 

5 

371068 

371253 

371437 

371622 

371806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

380030 

181 

240 

380211 

3S0392 

380573 

380754 

380934 

381115 

3S1296 

381476 

381656 

381837 

181 

] 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

160 

2 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

179 

3 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

4 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

5 

9166 

9343 

9520 

9698 

9875 

390051 

390228 

390405 

390582 

390759 

177 

6 

390935 

391112 

391288 

391464 

391641 

1817 

1993 

2169 

2345 

2521 

176 

7 

2697 

2873 

304S 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

8 

4452 

4627 

4S02 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175 

9 

6199 

6374 

6543 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

397940 

398114 

398237 

398461 

39S634 

398808 

398981 

399154 

399328 

399501 

173 

1 

9674 

9S47 

400020 

400192 

400365 

400538 

400711 

400883 

401056 

401228 

173 

2 

401401 

401573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

3 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

171 

4 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171 

5 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169 

7 

9933 

410102 

410271 

410440 

410609 

410777 

410946 

411114 

411283 

411451 

169 

8 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

9 

3300 

3467 

3635 

3303 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

414973 

415140 

415307 

415474 

415641 

415808 

415974 

416141 

416308 

416474 

167 

1 

6641 

6307 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

2 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

3 

9956 

420121 

4202S6 

420451 

420616 

420781 

420945 

421110 

421275 

421439 

165 

4 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

164 

5 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4882 

5045 

5208 

5371 

5534 

5697 

5860 

6023 

6186 

6349 

163 

7 

6511 

6674 

6S36 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

162 

8 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

9 

9752 

9914 

430075 

430236 

430398 

430559 

430720 

430881 

431042 

431203 

161 

270 

431364 

431525 

431685 

431846 

432007 

432167 

432328 

432488 

432649 

432809 

161 

1 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

2 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

159 

3 

6163 

6322 

6481 

6640 

6799 

6957 

7116 

7275 

7433 

7592 

159 

4 

7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

5 

9333 

9491 

9648 

9806 

9964 

440122 

440279 

440437 

440594 

440752 

158 

6 

440909 

441066 

441224 

441381 

441538 

1695 

1852 

2009 

2166 

2323 

157 

7 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

157 

8 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

156 

9 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

No. 

1 o 

1 

O 

3 

41 

5 

G 

7 

8 

9 

Biff. 






























TABLE XII. LOGARITHMS OF NUMBERS. 


159 


No 

0 

1 

2 

3 

4r 

5 

G 

7 

8 

9 

Diff. 

230 

147158 

447313 

447463 

447623 

447778 

447933 

448038 

448242 

448397 

448552 

155 

1 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

450095 

154 

2 

150219 

450403 

450557 

450711 

450865 

451018 

451172 

451326 

451479 

1633 

154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

152 

6 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

152 

7 

7882 

8033 

8184 

8336 

8437 

8638 

8789 

8940 

9091 

9242 

151 

8 

9392 

9543 

9694 

9845 

9995 

460146 

460296 

460447 

460597 

460748 

151 

9 

460898 

461048 

461193 

461348 

461499 

1649 

1799 

1948 

209S 

2248 

150 

290 

162398 

462548 

462697 

462847 

462997 

463146 

463296 

463445 

463594 

463744 

150 

1 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

5333 

5532 

5630 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6363 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

5 

9322 

9969 

470116 

470263 

470410 

470557 

470701 

470851 

470998 

471145 

147 

6 

471292 

471438 

1535 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4503 

4653 

4799 

4944 

5090 

5235 

5331 

5526 

146 

9 

5671 

5316 

5962 

6107 

6252 

6397 

6542 

6687 

6S32 

6976 

145 

300 

477121 

477266 

477411 

477555 

477700 

477844 

477939 

478133 

478278 

478422 

145 

1 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

2 

480007 

480151 

480294 

4S0433 

480582 

480725 

480369 

481012 

481156 

481299 

144 

3 

1443 

1536 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

4 

2374 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

5 

4300 

4442 

4535 

4727 

4869 

5011 

5153 

5295 

5437 

5579 

142 

6 

5721 

5363 

6005 

6147 

6289 

6430 

6572 

6714 

6355 

6997 

142 

7 

7133 

7230 

7421 

7583 

7704 

7845 

7986 

8127 

8269 

8410 

141 

8 

8551 

8692 

8333 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

9 

9953 

490099 

490239 

490330 

490520 

490661 

490301 

490941 

491081 

491222 

140 

310 

491362 

491502 

191642 

4917S2 

491922 

492062 

492201 

492341 

492431 

492621 

140 

1 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

2 

4155 

4294 

4433 

4572 

4711 

4850 

4939 

5128 

5267 

5406 

139 

3 

5544 

5633 

5822 

5960 

6099 

6233 

6376 

6515 

6653 

6791 

139 

4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

5 

8311 

8443 

8536 

8724 

8362 

8999 

9137 

9275 

9412 

9550 

138 

6 

9637 

9824 

9962 

500099 

500236 

500374 

500511 

500648 

500785 

500922 

137 

7 

501059 

501196 

501333 

1470 

1607 

1744 

1830 

2017 

2154 

2291 

137 

8 

2427 

2564 

2700 

2337 

2973 

3109 

3246 

3332 

3518 

3655 

136 

9 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

5014 

136 

320 

505150 

505286 

505421 

505557 

505693 

505828 

505964 

506099 

506234 

506370 

136 

1 

6505 

66 40 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

2 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

3 

9203 

9337 

9471 

9606 

9740 

9874 

510009 

510143 

510277 

510411 

134 

4 

510545 

510679 

510313 

510947 

511081 

511215 

1349 

1482 

1616 

1750 

134 

5 

1833 

2017 

2151 

2284 

2418 

2551 

2634 

2318 

2951 

3034 

133 

6 

3218 

3351 

3434 

3617 

3750 

3883 

4016 

4149 

4282 

4415 

133 

. 7 

4543 

4681 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

133 

8 

5374 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

132 

9 

7196 

7323 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

518514 

518646 

518777 

518909 

519040 

519171 

519303 

519434 

519566 

519697 

131 

I 

9328 

9959 

520090 

520221 

520353 

520434 

520615 

520745 

520376 

521007 

131 

2 

521133 

521269 

1400 

1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

■3 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

4 

3746 

3376 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

5 

5045 

5174 

5301 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

6 

6339 

6469 

6593 

6727 

6356 

6985 

7114 

7243 

7372 

7501 

129 

1 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

129 

e 

S917 

9045 

9174 

9302 

9130 

9559 

9687 

93.15 

9943 

530072 

128 

9 

530200 

53032- 

530456 

53058-1 

530712 

530340 

530968 

531096 

531223 

1351 

12S 

| No 
u 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

Diff. 
































































TABLE XII. LOGARITHMS OF .NUMBERS 


160 


No. 

0 

1 1 

2 

3 

4 

5 

G 

7 

8 

9 

Diff. 

340 

531479 

531607 

531734 

531862 

531990 

532117 

532245 

532372 

532500 

532627 

128 

1 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

2 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

3 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

4 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

5 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

6 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

540079 

540204 

125 

7 

540329 

540455 

540580 

540705 

540830 

540955 

541080 

541205 

1330 

1454 

125 

8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

9 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

544068 

544192 

544316 

544440 

544564 

544688 

544812 

544936 

545060 

545133 

124 

1 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

2 

6543 

6666 

67S9 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

3 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

550106 

123 

5 

550228 

550351 

550473 

550595 

550717 

550840 

550962 

551084 

551206 

1328 

122 

6 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

7 

2663 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

8 

3383 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

9 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

556303 

556423 

556544 

556664 

556785 

556905 

557026 

557146 

557267 

557387 

120 

1 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

2 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

3 

9907 

560026 

560146 

560265 

560385 

560504 

560624 

560743 

560863 

560982 

119 

4 

581101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

5 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

6 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

7 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118 

8 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

9 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

56S202 

563319 

568436 

568554 

568671 

568788 

56S905 

569023 

569140 

569257 

117 

1 

9374 

9491 

9608 

9725 

9842 

9959 

570076 

570193 

570309 

570426 

117 

2 

570543 

570660 

570776 

570893 

571010 

571126 

1243 

1359 

1476 

1592 

117 

3 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

4 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116 

5 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

6 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

7 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

8 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

115 

9 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

330 

579784 

579S98 

580012 

580126 

580241 

580355 

580469 

580583 

580697 

580811 

114 

] 

580925 

581039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

2 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

3 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4 105 

4218 

113 

4 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113 

5 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

6 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

llfi 

- 7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

860S 

8720 

112 

8 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

1)2 

9 

9950 

590061 

590173 

590284 

590396 

590507 

590619 

590730 

590S42 

590953 

112 

390 

591065 

591176 

5912S7 

591399 591510 

591621 

591732 

591843 

591955 

592066 

111 

1 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

2 

3236 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

111 

3 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

110 

4 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

5 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

6 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

7 

8791 

8900 

900S 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

109 

8 

9883 

9992 

600101 

600210 600319 

600428 

*600537 

600646 600755 

600S64 

109 

E 

60097C 

601082 

1191 

129S 

| 1408 

1517 

1625 

1734 

1843 

1951 

109 

No 

0 

} 1 

2 

3 

1 4 

5 

1 6 

7 

1 8 

1 9 

Biff 

_jL 






















































TABLE XII. LOGARITHMS OF NUMBERS 


161 


| No. 

0 

1 

2 1 

3 

4 

5 

6 

7 

8 

9 

Diff 1 

400 

302060 

602169 

602277 

6023S6 

602494 

602603 

602711 

602819 

602928 

603036 

108 

1 

3144 

3253 

3361 

3469 

3577 

3636 

3794 

3902 

4010 

4118 

103! 

2 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

108 

3 

5305 

541.3 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

108 

4 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

107 

5! 

7455 

7562 

7669 

7777 

7884 

7991 

8093 

8205 

8312 

8419 

107 

6i 

8526 

8633 

8740 

8347 

8954 

9061 

9167 

9274 

9381 

9488 

107 

7 

9594 

9701 

9308 

99141610021 

610123 

610234 

610341 

610447 

610554 

107 

8 

610660 

610767 

610873 1 

610979 

1086 

1192 

1298 

1405 

1511 

1617 

106 

9 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

612784 

612390 

612996 

613102 

613207 

613313 

613419 

613525 

613630 

613736 

106 

1 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

106 

2 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

5845 

105 

3 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6636 

6790 

6S95 

105 

4 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

105 

5 

8043 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8834 

8989 

105 

6 

9093 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

620032 

104 

7 

620136 

620240 

620344 

620448 

620552 

620656 

620760 

620864 

620968 

1072 

104 

8 

1176 

1280 

1334 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

104 

9 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

104 

420 

623249 

623353 

623456 

623559 

623663 

623766 

623369 

623973 

624076 

624179 

103 

I 

4232 

4385 

4488 

4591 

4695 

4793 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

103 

3 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

103 

4 

7366 

7463 

7571 

7673 

7775 

7878 

7930 

8082 

8185 

8287 

102 

5 

8339 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

6 

9410 

9512 

9613 

9715 

9817 

9919 

630021 

630123 

630224 

630326 

102 

7 

630428 

630530 

630631 

630733 

630835 

630936 

1038 

1139 

1241 

1342 

102 

8 

1444 

1545 

1647 

1748 

1S49 

1951 

2052 

2153 

2255 

2356 

101 

9 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

101 

430 

63346S 

633569 

633670 

633771 

633372 

633973 

634074 

634175 

634276 

634376 

101 

1 

4477 

4578 

4679 

4779 

4330 

4931 

5031 

5182 

5283 

5333 

101 

2 

5484 

5584 

5685 

5785 

5836 

5986 

6087 

6187 

6287 

6338 

100 

3 

648S 

6533 

6688 

6789 

6389 

6939 

7089 

7189 

7290 

7390 

100 

4 

7490 

7590 

7690 

-7790 

7890 

7990 

8090 

8190 

8290 

8389 

100 

5 

8489 

8589 

8639 

8739 

8888 

8988 

9038 

9183 

9287 

9387 

100 

6 

9436 

9586 

9636 

9785 

98S5 

9984 

640034 

640183 

640233 

640382 

99 

7 

640481 

640581 

640630 

640779 

640879 

640978 

1077 

1177 

1276 

1375 

99 

8 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

9 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440 

643453 

643551 

643650 

643749 

643847 

643946 

644044 

644143 

644242 

644340 

98 

1 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

9S 

2 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306. 

93 

3 

6404 

6502 

6600 

6693 

6796 

6894 

6992 

7089 

7187 

7285 

98 

4 

7333 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165' 

8262 

98 

' 5 

8360 

8458 

8555 

8653 

8750 

8348 

8945 

9043 

9140 

9237 

97 

, G 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

650016 

650113 

650210 

97 

' 7 

650303 

650405 

650502 

650599 

650696 

650793 

650890 

0987 

1084 

1181 

97 

8 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

9 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

450 

653213 

653309 

653405 

653502 

653598 

653695 

653791 

6538S8 

653984 

654080 

96 

1 

4177 

4273 

4369 

4465 

4562 

4653 

4754 

4850 

4946 

5042 

96 

2 

5133 

5235 

5331 

5427 

5523 

5619 

5715 

5810 

5906 

6002 

96 

3 

6093 

6194 

6290 

6336 

6482 

6577 

6673 

6769 

6864 

6960 

96 

4 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

96 

5 

8011 

8107 

8202 

8293 

8393 

8483 

8584 

8679 

8774 

8870 

95 

6 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

9S21 

95 

7 

9916 

660011 

660106 

660201 

660296 

660391 

660486 

660581 

660676 

660771 

95 

8 

660365 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

95 

9 

1813 

1907 

2002 

2096 

2191 

2236 

2380 

2475 

2569 

2663 

95 

No 

0 

1 1 

2 

3 

4 

5 

6 

7 

8 

9 

BifT. 




































































162 


TABLE XII. LOGARITHMS OF NUMBERS 


— 

! 6 

0 

1 1 

2 

3 

I 

5 

G 

7 

§ 


Diff. ) 

46J 

662758 

662352 

662947 

663041 

663135 

663230 

663324 

663418 

663512:663607 

94 

1 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

94 

2 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

5581 

5675 

5769 

5862 

5956 

6050 

6143 

6237 

6331 

6424 

94 

4 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

94 

5 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

93 

6 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

7 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

670060 

670153 

93 

8 

670246 

670339 

670431 

670524 

670617 

670710 

670802 

670895 

0988 

1080 

93 

9 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

672190 

672283 

672375 

672467 

672560 

672652 

672744 

672836 

672929 

92 

1 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3350 

92 

2 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

45S6 

4677 

4769 

92 

3 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

92 

4 

5778 

5S70 

5S62 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

92 

5 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

91 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

91 

7 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

91 

8 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

680063 

680154 

680245 

91 

9 

680336 

680426 

680517 

680607 

680698 

680789 

680S79 

0970 

1060 

1151 

91 

480 

681241 

681332 

6S1422 

681513 

681603 

631693 

681784 

681874 

681964 

682055 

90 

1 

2145 

2235 

2326 

2416 

2506 

2598 

2686 

2777 

2867 

2957 

90 

2 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

3 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

4 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

90 

5 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

6 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

7 

7529 

7618 

7707 

7798 

7886 

7975 

8064 

8153 

8242 

8331 

89 

8 

8420 

S509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

89 

9 

9309 

9393 

9486 

9575 

9664 

9753 

9841 

9930 

690019 

690107 

89 

490 

690196 

690285 

690373 

690462 

690550 

690639 

690728 

690816 

690905 

690993 

89 

1 

1031 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

88 

2 

1965 

2053 

2142 

2230 

2318 

2405 

2494 

2583 

2671 

2759 

88 

3 

2347 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

4 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

5 

4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

83 

6 

5432 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

7 

6356 

6444 

6531 

661S 

6706 

6793 

6880 

6963 

7055 

7142 

87 

8 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

• 9 

8101 

S1S8 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

8883 

87 

500 

698970 

699057 

699144 

699231 

699317 

699404 

699491 

699578 

699664 

699751 

87 

] 

9333 

9924 

700011 

700098 

700184 

700271 

700358 

700444 

700531 

700617 

87 

2 

700704 

700790 

0377 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

86 

n 

O 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

4 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

86 

5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4403 

4494 

4579 

4665 

4751 

4837 

4922 

86 

7 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

86 

8 

5364 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

9 

6718 

6303 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

85 

510 

707570 

707655 

707740 

707826 

707911 

707996 

708081 

708166 

708251 

708336 

85 

1 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

85 

2 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

710033 

85 

3 

710117 

710202 

710287 

710371 

710156 

710540 

710625 

710710 

710794 

0879 

85 

4 

0963 

1048 

1132 

1217 

1 1301 

1385 

1470 

1554 

1639 

1723 

84- 

5 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

84 

6 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3323 

3107 

84 

7 

3491 

3575 

3659 

3742 

3326 

3910 

3994 

4078 

4162 

4246 

84 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4333 

4916 

5000 

5084 

84 

9 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

84 

! No 

0 

1 

2 

3 

1 4 

5 

G 

7 

8 

9 

Diff. 























































TABLE XII 


LOGARITHMS OF NUMBERS 


163 


No.l 

0 

s 

a | 

3 | 

4r 

5 

6 

7 1 

8 

9 

Diff. 

520 

r 16003 

716037 

716170 

716254 

716337 

716421 

716504 

716583 

716671 

716754 

83 

1 

6838 

6921 

7004 

7038 

7171 

7254 

7333 

7421 

7504 

7587 

83 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

83 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9165 

9248 

83 

4 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

720077 

83 

5 

720159 

720242 

720325 

720407 

720490 

720573 

720655 

720733 

720321 

0903 

83 

G 

0986 

1063 

1151 

1233 

1316 

1398 

1431 

1563 

1646 

1728 

82 

7 

1811 

1893 

1975 

2053 

2140 

2222 

2305 

2337 

2469 

2552 

82 

8 

2634 

2716 

2793 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

1 gl 
J 1 

3456 

3538 

362'! 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530:724276 

72435S 

724440 

724522 

724604 

724635 

724767 

724849 

724931 

725013 

82 

1 

5095 

5176 

5253 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

82 

2 

5912 

5993 

6075 

6158 

6233 

6320 

6401 

6483 

6564 

6616 

82 

O 

O 

6727 

6309 

6390 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

6 

9165 

9246 

9327 

9403 

9489 

9570 

9651 

9732 

9313 

9893 

81 

7 

9974 

730055 

730136 

730217 

730293 

730378 

730459 

730540 

730621 

730702 

81 

8 

7307S2 

0363 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

9 

1539 

1669 

1750 

1330 

1911 

1991 

2072 

2152 

2233 

2313 

81 

540 

732394 

732474 

732555 

732635 

732715 

732796 

732376 

732956 

733037 

733117 

80 

I 

3197 

3278 

3353 

3138 

3518 

3598 

3679 

3759 

3839 

3919 

80 

2 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

80 

3 

4800 

4830 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

5519 

80 

4 

5599 

5679 

5759 

5833 

5918 

5998 

6078 

6157 

6237 

6317 

80 

5 

6397 

6476 

6556 

6635 

6715 

6795 

6374 

6954 

7034 

7113 

80 

6 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

79 

7 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

8 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

9 

9572 

9651 

9731 

9310 

9889 

9968 

740047 

740126 

740205 

740234 

79 

550 

740363 

740442 

710521 

740600 

740678 

740757 

740836 

740915 

740994 

741073 

79 

1 

1152 

1230 

1309 

1383 

1467 

1546 

1624 

1703 

1782 

1860 

79 

2 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

24S9 

2568 

2647 

79 

3 

2725 

2804 

2382 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

4 

3510 

3538 

3667 

3745 

3323 

3902 

3980 

4058 

4136 

4215 

78 

5 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

6 

5075 

5153 

5231 

5309 

5337 

5465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933 

6011 

6039 

6167 

6245 

6323 

6401 

6479 

6556 

78 

8 

6634 

6712 

6790 

6363 

6945 

7023 

7101 

7179 

7256 

7334 

78 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

560 

748183 

743266 

743343 

748421 

743498 

748576 

748653 

748731 

748808 

748885 

77 

1 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

77 

2 

9736 

9814 

9891 

9968 

750045 

750123 

750200 

750277 

750354 

750431 

77 

3 

750503 

750586 

750663 

750740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

4 

1279 

1356 

1433 

1510 

1587 

1664 

1741 

1818 

1S95 

1972 

77 

5 

2043 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

77 

6 

2316 

2393 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

77 

7 

3533 

3650 

3736 

3813 

3389 

3966 

4042 

4119 

4195 

4272 

77 

8 

4348 

4425 

4501 

4578 

4654 

4730 

4807 

4383 

4960 

5036 

76 

9 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

76 

570 

755375 

753951 

756027 

756103 

756180 

756256 

756332 

756108 

756484 

756560 

76 

l 

6836 

6712 

6788 

6364 

6940 

7016 

7092 

7163 

7244 

7320 

76 

2 

7396 

7472 

7548 

7621 

7700 

7775 

7851 

7927 

8003 

8079 

76 

3 

8155 

8230 

8306 

8332 

8458 

8533 

8609 

8685 

8761 

8836 

76 

4 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

76 

5 

9668 

9743 

9819 

9394 

9970 

760045 

760121 

760196 

760272 

760347 

75 

G 

760422 

760493 

760573 

760649 

760724 

0799 

0375 

0950 

1025 

1101 

75 

. 7 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

75 

8 

1926 

2003 

2073 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

75 

9 

2679 

2754 

2329 

2904 

2978 

3053 

3123 

3203 

3278 

3353 

75 

No 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dur. 























































164 


TABLE XII. LOGARITHMS OF NUMBERS 


ry 

No. 

0 

1 

2 i 

3 

4 

5 

G 

7 

8 

9 

Diff. 

5S0 

763428 

763503 

763578 

763653 

763727 

763802 

763877 

763952 

764027 

764101 

75 

1 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

75 

2 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

75 

3 

5669 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

74 

4 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

74 

5 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

6 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

74 

8 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

770042 

74 

9 

770115 

770189 

770263 

770336 

770410 

770484 

770557 

770631 

770705 

0778 

74 

590 

770852 

770926 

770999 

771073 

771146 

771220 

771293 

771367 

771440 

771514 

74 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

2 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

48S2 

4955 

5028 

5100 

5173 

73 

6 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

73 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

73 

8 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

600 

778151 

778224 

778296 

778368 

778441 

778513 

778585 

778658 

778730 

778802 

72 

] 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

2 

9596 

9669 

9741 

9813 

9885 

9957 

780029 

780101 

780173 

780245 

72 

3 

780317 

780389 

780461 

780533 

780605 

780677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

13% 

1468 

1540 

1612 

1684 

72 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

785401 

785472 

785543 

785615 

785686 

785757 

785828 

785899 

785970 

71 

1 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

71 

3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

71 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

71 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

6 

9581 

9651 

9722 

9792 

9863 

9933 

790004 

790074 

790144 

790215 

70 

7 

790285 

790356 

790426 

790496 

790567 

790637 

0707 

0778 

0848 

0918 

70 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

792462 

792532 

792602 

792672 

792742 

792812 

792982 

792952 

793022 

70 

1 

3092 

3162 

3231 

330 r 

3371 

3441 

3511 

3581 

3651 

3721 

70 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

70 

3 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

70 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

70 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

69 

6 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

69 

7 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752 

7821 

7890 

69 

8 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69 

9 

S651 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

799341 

799409 

799478 

799547 

799616 

799685 

799754 

799823 

799892 

799961 

69 

1 

800029 

800098 

800167 

800236 800305 

800373 

800442 

800511 

800580 

800648 

69 

2 

0717 

0786 

0S54 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

69 

3 

1404 

1472 

1541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

69 

4 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

G3 

5 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

68 

6 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

68 

7 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

68 

c 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

c 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

68 

No 

0 

• 1 

2 

3 

1 4 

5 

6 

7 

8 

9 

DifT. 
















































TABLE XII. LOGARITHMS OF NUMBERS. 


H>5 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

BiJLj 

640 

306180 

806218 

806316 

303334 

806451 

806519 

806587 

806655 

806723 

806790 

6 $ 

1 

6358 

69 6 

6994 

7061 

7129 

7197 

7264 

7332 

7400 

7467 

68 

2 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

68 

3 

8211 

8279 

8346 

8-414 

8481 

8549 

861 o 

8684 

8751 

8818 

67 

4 

8386 

8953 

9021 

9083 

9156 

9223 

9290 

9358 

9425 

9492 

67 

5 

9560 

9827 

9694 

9762 

9329 

9896 

9964 

810031 

810098 

810165 

67 

6 

810233 

810300 

810367 

810434 

810501 

810569 

810636 

0703 

0770 

0837 

67 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

9 

2245 

2312 

2379 

2145 

2512 

2579 

2646 

2713 

2780 

2847 

67 

650 

812913 

812980 

813047 

813114 

813181 

813247 

813314 

813381 

813448 

813514 

67 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

66 

4 

5578 

5644 

5711 

5777 

5S43 

5910 

5976 

6042 

6109 

6175 

66 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

66 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

7 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

6S 

9 

8885 

8951 

9017 

9033 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819544 

819610 

819676 

819741 

819S07 

819S73 

319939 

320004 

820070 

820136 

66 

1 

820201 

820267 

820333 

820399 

320464 

820530 

820595 

0661 

0727 

0792 

66 

2 

0858 

0924 

0939 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

65 

5 

2822 

2837 

2952 

3018 

. 3083 

3143 

3213 

3279 

3344 

3409 

65 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

65 

7 

4126 

4191 

4256 

4321 

4336 

4451 

4516 

4581 

4646 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

65 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5S80 

5945 

6010 

65 

670 

826075 

826140 

826204 

826269 

826334 

826399 

826464 

826528 

826593 

826658 

65 

1 

6723 

6787 

6352 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

65 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

3 

8015 

8080 

8144 

8209 

8273 

8333 

8402 

8467 

8531 

8595 

64 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

5 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

64 

6 

9947 

830011 

830075 

830139 

830204 

830263 

830332 

830396 

830460 

830525 

64 

7 

830539 

0653 

0717 

0731 

0845 

0909 

0973 

1037 

1102 

1166 

64 

8 

1230 

1294 

1358 

1422 

I486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1993 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

6SC 

832509 

832573 

832637 

832700 

832764 

832828 

832392 

832956 

833020 

833083 

64 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

64 

2 

3784 

3348 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

4 

5056 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

63 

5 

5691 

5754 

5317 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6S30 

6894 

63 

7 

6957 

7020 

7033 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

8 

7588 

7652 

7715 

7773 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

87S6 

63 

690 

833849 

838912 

833975 

839033 

839101 

839164 

839227 

839289 

839352 

S39415 

63 

1 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

840043 

63 

2 

840106 

840169 

840232 

840294 

840357 

840420 

840482 

840545 

840608 

0671 

63 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

63 

5 

1935 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

62 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

62 

7 

3233 

3295 

3357 

3120 

3482 

3544 

3606 

3669 

3731 

3793 

62 

S 

3855 

3918 

3930 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

62 

S 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

62 

No 

0 

| 1 

2 

3 

4: 

5 

6 

7 

8 

9 

Diff. 































































l'ABLE XII. 


LOGARITHMS OF NUMBERS. 


166 


No. 

0 

1 

a 

3 

4r 

5 

G 

7 

8 

» 

Dill. 

700 

845098 

845160 

845222 

845284 

845346 

8454OS 

845470 

845532 

8455S4 

845656 

62 

1 

5713 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

62 

2 

6337 

6399 

6461 

6523 

6585 

66-16 

6708 

6770 

6832 

6S94 

62 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

621 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

62 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

62 . 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

8 

850033 

850095 

850156 

850217 

850279 

850340 

850401 

850462 

850524 

850585 

61 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

851258 

851320 

851381 

851442 

851503 

851564 

851625 

851686 

851747 

851809 

61 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

4 

3698 

3759 

3S20 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

61 

• 5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

857393 

857453 

857513 

857574 

857634 

S57694 

857755 

857815 

857875 

6 C 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

860038 

860098 

860158 

860218. 

S60278 

60 

5 

860338 

860398 

860458 

860518 

860578 

0637 

0697 

0757 

0817 

0877 

60 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

60 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

* 3204 

3263 

60 

730 

863323 

863382 

863442 

863501 

863561 

863620 

8636S0 

863739 

863799 

863858 

59 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4S67 

4926 

4985 

5045 

59 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

59 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

59 

5 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

59 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

59 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

869232 

869290 

869349 

869408 

869466 

869525 

869584 

869642 

869701 

869760 

59 

1 

9818 

9877 

9935 

9994 

870053 

870111 

870170 

870228 

870287 

S70345 

59 

2 

870404 

870462 870521 

870579 

0638 

0696 

0755 

0813 

0872 

0930 

58 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

58 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

58 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

68 

750 

875061 

875119 

875177 

875235 

875293 

875351 

875409 

875466 

875524 

875582 

58 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

2 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

5S 

3 

6795 

6353 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

5 

7947 

8004 

i 8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57 

6 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

7 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

8 

9669 

9726 

9784 

9841 

9898 

9956 

880013 

880070 

880127 

880185 

57 

9 

880242 

880299 880356 

880413 

880471 

880528 

0585 

0642 

0699 

0756 

57 

•No 

i 1 - 

0 

1 

I 3 

3 

4 

5 

G 

7 

8 

9 

DiSF. 








































TABLE XII. LOGARITHMS OF NUMBERS 


167 


r- 

No. 

0 

1 

2 

3 

4: 

5 

G 

7 

8 

9 

Diff. 

760 

880814 

880371 

880928 

880985 

881042 

381099 

881156 

881213 

881271 

881328 

57 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

57 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

. 3491 

3548 

3605 

57 

5 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

57 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4632 

4739 

57 

7 

4795 

4352 

4909 

4965 

5022 

5078 

5135 

5192 

5243 

5305 

57 

8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

9 

5926 

5933 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

56 

770 

836491 

8S6547 

886604 

886660 

886716 

886773 

886829 

886335 

8S6942 

886998 

56 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

56 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

56 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

56 

4 

8741 

8797 

8353 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

56 

5 

9302 

9358 

9414 

9470 

9526 

9582 

9633 

9694 

9750 

9806 

56 

6 

9362 

9913 

9974 

890030 

890086 

890141 

890197 

890253 

890309 

890365 

56 

7 

890421 

390477 

890533 

05S9 

0645 

0700 

0756 

0812 

0 S68 

0924 

56 

8 

0930 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

56 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

192S 

1983 

2039 

56 

780 

892095 

892150 

S92206 

892262 

392317 

892373 

892429 

892484 

892540 

892595 

56 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

56 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

3 

3762 

3317 

3373 

3928 

' 3984 

4039 

4094 

4150 

4205 

4261 

55 

4 

4316 

4371 

4427 

4432 

4538 

4593 

4648 

4704 

4759 

4814 

55 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

55 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

55 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

55 

8 

6526 

6581 

6636 

6692 

6747 

6302 

6357 

6912 

6967 

7022 

55 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

897527 

8976S2 

897737 

897792 

897847 

897902 

897957 

893012 

898067 

898122 

55 

l 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

55 

2 

8725 

8780 

8335 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

3 

9273 

9323 

9333 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

4 

9321 

9375 

9930 

9935 

900039 

900094 

900149 

900203 

900258 

900312 

55 

5 

900367 

900422 

900476 

900531 

0586 

0640 

0695 

0749 

0804 

0859 

55 

6 

0913 

0963 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

54 

9 

2547 

2601 

2655 

2710 

2764 

2318 

2873 

2927 

2981 

3036 

54 

800 

003090 

903144 

903199 

9032(53 

903307 

903361 

903416 

903470 

903524 

903578 

54 

1 

3633 

3687 

3741 

3795 

3349 

3904 

3958 

4012 

4066 

4120 

54 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

3 

4716 

4770 

4321 

4878 

4932 

4936 

5040 

5094 

5143 

5202 

54 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

6 

6335 

6339 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

54 

7 

6874 

6927 

6981 

7035 

7039 

7143 

7196 

7250 

7304 

7358 

54 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

9 

7949 

8002 

8056 

8110 

8163 

8217 

, 8270 

8324 

8378 

8431 

54 

810 

908485 

903539 

908592 

908616 

908699 

90S753 

90SS07 

903860 

90S914 

903967 

54 

1 

9021 

9074 

9123 

9131 

9235 

9289 

9342 

9396 

9449 

9503 

54 

2 

9556 

9610 

9663 

9716 

9770 

9323 

9877 

9930 

99S4 

910037 

53 

3 

910091 

910144 

910197 

910251 

910304 

910358 

910411 

910464 

910518 

0571 

53 

4 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

53 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

53 

6 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

53 

7 

2222 

2275 

2323 

2331 

2435 

2488 

2541 

2594 

2647 

2700 

53 

8 

2753 

2306 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

53 

9 

3234 

3337 

3390 

3143 

3496 

3549 

3602 

3655 

3708 

3761 

53 

No. 

li -— 

0 

1 

2 

3 

4L 

5 

6 

7 

8 

9 

Diff. 

































168 


TABLE XII. LOGARITHMS OF NUMBERS 


No. 

0 

1 

a 

3 

4 

5 

G 

7 

8 

9 

Diff. 

820 

913314 

913867 

913920 

913973 

914026 

914079 

914132 

914184 

914237 

914290 

53 

1 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

53 

2 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

53 

3 

6400 

5453 

5505 

5558 

5611 

5664 

5716 

6769 

5822 

5875 

53 

4 

6927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

5 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

6 

6980 

7033 

7085 

7133 

7190 

7243 

7295 

7348 

7400 

7453 

53 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

919078 

919130 

919183 

919235 

919287 

919340 

919392 

919444 

919496 

919549 

52 

1 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

920019 

920071 

52 

2 

920123 

920176 

920228 

920280 

920332 

920384 

920436 

920489 

0541 

0593 

52 

3 

0645 

0697 

0749 

0S01 

0853 

0906 

0958 

1010 

1062 

1114 

52 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

52 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

52 

6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

52 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

52 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

52 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

924279 

924331 

9243S3 

924434 

924486 

924538 

924589 

924641 

924693 

924744 

52 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

62 

2 

5312 

5364 

5415 

5467 

5518 

5570 

- 5621 

5673 

5725 

5776 

62 

3 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

51 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

61 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

51 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

61 

7 

7833 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

51 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

51 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

51 

850 

929419 

929470 

929521 

929572 

929623 

929674 

929725 

929776 

929S27 

929879 

51 

J 

9930 

9931 

930032 

930083 

930134 

930185 

930236 

930287 

930338 

930389 

51 

2 

930440 

930491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

51 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

51 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

5 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

51 

7 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

51 

8 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

51 

860 

934498 

934549 

934599 

934650 

934700 

934751 

934801 

934852 

934902 

934953 

50 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

50 

2 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

50 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

50 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

50 

5 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

S 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

50 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

[870 

939519 

939569 

939619 

939669 

939719 

939769 

939819 

939869 

939918 

939968 

50 

1 

940018 

940063 

940118 

940168 

940218 

940267 

940317 

940367 

940417 

940467 

50 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

50 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

50 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

50 

6 

2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

50 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

49 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

9 

3939 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

49 

No 

0 

1 

3 

3 

4 

5 

6 

r 

8 

9 

Diff. 












































TABLE XII. LOGARITHMS OF NUMBERS 


169 


No. 

O 

i 1 

2 

3 

4r 

5 

6 

7 

8 

9 

Diff. 

SSO 

944483 

944532 

944581 

944631 

944630 

944729 

944779 

944328 

944877 

944927 

49 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

49 

2 

6069 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5362 

5912 

49 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

49 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6S94 

49 

5 

6943 

6992 

7041 

7090 

7140 

7189 

7233 

7287 

7336 

7385 

49 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

8 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8S53 

49 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

S90 

949390 

949439 

94948S 

949536 

949585 

949634 

949683 

949731 

949780 

949829 

49 

I 

9878 

9926 

9975 

950024 

950073 

950121 

950170 

950219 

950267 

950316 

49 

2 

950365 

950414 

950462 

0511 

0560 

0603 

0657 

0706 

0754 

0803 

49 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

49 

4 

1338 

1336 

1435 

1483 

1532 

1580 

1829 

1677 

1726 

1775 

49 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

48 

7 

2792 

2841 

2889 

2933 

2986 

3034 

3083 

3131 

3180 

3228 

48 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

48 

9 

3760 

3303 

3356 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

954291 

954339 

954337 

954435 

954484 

954532 

954580 

954628 

954677 

48 

1 

4725 

4773 

4821 

4369 

4918 

4968 

5014 

5062 

5110 

5158 

48 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

3 

5688 

5736 

5784 

5832 

5380 

5928 

5976 

6024 

6072 

6120 

48 

4 

6168 

6216 

6265 

6313 

6361 

6109 

6457 

6505 

6553 

6601 

48 

5 

6649 

6697 

6745 

6793 

6840 

6388 

6936 

6984 

7032 

7080 

48 

6 

7123 

7176 

7224 

7272 

7320 

7363 

7416 

7464 

7512 

7559 

48 

7 

7607 

7655 

7703 

7751 

7799 

7347 

7894 

7942 

7990 

8038 

48 

8 

8036 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

,9 

8564 

8612 

8659 

8707 

8755 

8803 

8350 

8898 

8946 

8994 

48 

910 

959041 

959089 

959137 

959185 

959232 

959230 

959328 

959375 

959423 

959471 

48 

] 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

48 

2 

9995 

960042 

960090 

960133 

960185 

960233 

960230 

960328 

960376 

960423 

48 

3 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0304 

0851 

0399 

48 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

48 

5 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

6 

1895 

1943 

1990 

2038 

2035 

2132 

2180 

2227 

2275 

2322 

47 

7 

2369 

2417 

2461 

2511 

2559 

2606 

2653 

2701 

2743 

2795 

47 

8 

2343 

2890 

2937 

2935 

3032 

3079 

3126 

3174 

3221 

3263 

47 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920 

96378S 

963835 

963882 

963929 

963977 

964024 

964071 

964118 

964165 

964212 

47 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

2 

4731 

4778 

4825 

4372 

4919 

4966 

5013 

5061 

5108 

5155 

47 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5434 

5531 

5578 

5625 

47 

4 

5672 

5719 

5766 

5313 

5860 

5907 

5954 

6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

47 

6 

6611 

6653 

6705 

6752 

6799 

6345 

6892 

6939 

6986 

7033 

47 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7403 

7454 

7501 

47 

8 

7548 

7595 

7642 

7638 

7735 

7782 

7829 

7875 

7922 

7969 

47 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47 

930 

963483 

968530 

963576 

968623 

963670 

963716 

968763 

968810 

968856 

968903 

47 

I 

8950 

.8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

47 

2 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47 

3 

9382 

9928 

9975 

970021 

970068 

970114 

970161 

970207 

970254 

970300 

47 

4 

970347 

970393 

970440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

46 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

46 

1 8 

2203 

2249 

2295 

2342 

2338 

2434 

2481 

2527 

2573 

2619 

46 

1 9 

2666 

2712 

2753 

2804 

2851 

2397 

2943 

2989 

3035 

3082 

46 

! No 

i - 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Biff. 




















































170 


TABLE XII. LOGARITHMS OF NUMBERS 


No. 

0 

1 

a 

3 

4 

1 5 

G 

7 

L 8 

1 9 

Dill. 

04 

973128 

973174 

973220 

973266 

973313 

973359 973405 

973451 973497 973543 

46 

1 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

46 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

3 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

46 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

5 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

6 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

8 

6S08 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

977724 

977769 

977815 

977861 

977906 

977952 

977998 

978043 

978089 

97$135 

46 

] 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

3 

9093 

913S 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

5 

980003 

980049 

980094 

980140 

980185 

980231 

980276 

980322 

980367 

980412 

15 

6 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

ose: 

45 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

982271 

982316 

982362 

982407 

982452 

982497 

982543 

982588 

982633 

982678 

45 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

45 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

40.32 

15 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

44^2 

45 

5 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

G 

4377 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5332 

45 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5*3'‘ 

45 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

936772 

936817 

986861 

986906 

986951 

9S6996 

987040 

987085 

987130 

987175 

45 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8063 

45 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

45 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

6 

9450 

9494 

9539 

9583 

9623 

9672 

9717 

9761 

9806 

9850 

44 

7 

9S95 

9939 

9983 

990028 

990072 

990117 

990161 

990206 

990250 

990294 

44 

8 

990339 

990333 

990428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

I LoZ 

44 

980 

991226 

991270 

991315 

991359 

991403 

991448 

991492 

991536 

991580 

991625 

44 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

44 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

295.1 

14 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3302 

44 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

14 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4?-3 

14 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

44 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

44 

990 

995635 

995679 

995723 

995767 

995811 

995S54 

995398 

995942 

995986 

996030 

44 

1 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6643 

66S7 

6731 

6774 

6818 

6S62 

69i A 

i4 

3 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

V343 

44 

4 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

5 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

82:8' 

*4 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

86o2 

44 

7 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

44 

8 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

95 f ->2 

44 

9 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

No. 

0 

1 

3 

3 

4 

5 

G 

7 

8 

9 

Diff. 

























































TABLE XIII. 


LOGARITHMIC SINES, COSINES, TANGENTS 


AND 


COTANGENTS. 


172 


TABLE XIII. LOGARITHMIC SINES, 


NOTE. 

Tiie table here given extends to minutes only. The usual method 
of extending such a table to seconds, by proportional parts of the 
difference between two consecutive logarithms, is accurate enough 
for most purposes, especially if the angle is not very small. When 
the angle is very small, and great accuracy is required, the following 
method may be used for sines, tangents, and cotangents. 

I. Suppose it were required to find the logarithmic sine of 5' 24". 
By the ordinary meth' v i we should have 

log. sin. 5' = 7.162696 

diff. for 24" = 31673 

log. sin. 5' 24" — 7.194369 

Tnc more accurate method is founded on the proposition in Trigo¬ 
nometry, that the sines or tangents of very small angles are propor¬ 
tional to the angles themselves. In the present case, therefore, we 
have sin. 5': sin. 5' 24' = 5': 5' 24'' = 300": 324". Hence sin. 5' 24' 

= “ ° , or log. sin. 5' 24" = log. sin. 5' + log. 324 — log. 300. 

The difference for 24" will therefore, be the difference between the 
logarithm of 324 and the logarithm of 3C0. The operation will stand 
thus : — 

log. 324 = 2.510545 

log. 300 = 2.477121 

diff. for 24 = 33424 

log. sin. 5' = 7.162696 

log. sin. 5' 24" = 7.196120 

Comparing this value with that given in tables that extend to seconds 
we find it exact even to the last figure 

II. Given log. sin. A = 7.004438 to find A. The sine next less 
than this in the table is sin. 3' = 6.940847. Now we have sin. 3': sin. A 

~ 3 • A. Therefore, A = , or log. A = log. 3 log. sin. A 

~ log. sin. 3'. Hence it appears, that, to find the logarithm of A nr 






COSINES, TANGENTS, AND COTANGENTS. 


173 


minutes, we must add to the logarithm of 3 the difference between 
iog. sin. A and log. sin. 3'. 

log. sin. A = 7.004438 
log. sin. 3' = 6.940847 

63591 

log. 3 = 0.477121 

A = 3.473 0.540712 

or A = S' 28.38". By the common method we should have found 
A — 3' 30.54". 

The same method applies to tangents and cotangents, except that in 
the case of cotangents the differences are to be subtracted. 


%* The radius of this table is unity, and the characteristics 9, 8, 7, 
and 6 stand respectively for — 1, — 2, —3, and —4. 





i74 TABLE XIII. LOGARITHMIC SINES, 


<P 


179^ 


M. 

Sine. 

D. 1 . 

Cosine. 

D. 1'. 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

Inf. nea. 

6.463726 
.764756 
.940847 

7.065786 

.162696 

.241877 

.308824 

.366316 

.417968 

7.463726 
.505118 
.542906 
.577668 
.609853 
.639316 
.667845 
.694173 
.718997 
.742478 

7.764754 

.785943 

.806146 

.825451 

.843934 

.861662 

.878695 

.895085 

.910879 

.926119 

7.940842 

.955082 

.963870 

.982233 

.995198 

8.007787 

.020021 

.031919 

.043501 

.054781 

8.065776 

.076500 

.086965 

.097183 

.107167 

.116926 

.126471 

.135810 

.144953 

.153907 

8.162681 

.171280 

.179713 

.187985 

.196102 

.204070 

.211895 

.219581 

.227134 

.234557 

.241855 

5017.17 
2934.85 
2082.31 

1615.17 
1319.69 
1115.78 

966.53 

852.54 
762.62 

689.88 

629.81 

579.37 

536.41 

499.38 

467.14 

438.81 

413.72 
391.35 
371.27 

353.15 

336.72 
321.75 
303.05 
295.47 

283.88 
273.17 

263.23 
253.99 

245.33 

237.33 

229.80 

222.73 
216.08 

209.81 
203.90 

198.31 
193.02 
188.01 
183.25 

178.72 

174.42 

170.31 

166.39 

162.65 
159.08 

155.66 
152.38 

149.24 
146.22 

143.33 

140.54 

137.86 
135.29 
132.80 
130.41 
128.10 

125.87 

123.72 
121.64 

0.000000 

.000000 

.000000 

.000000 

.000000 

.000000 

9.999999 

.999999 

.999999 

.999999 

9.999998 

.999998 

.999997 

.999997 

.999996 

.999996 

.999995 

.999995 

.999994 

.999993 

9.999993 

.999992 

.999991 

.999990 

.999989 

.999989 

.99998S 

.999987 

.999986 

.999985 

9.999983 

.999982 

.999931 

.9999S0 

.999979 

.999977 

.999976 

.999975 

.999973 

.999972 

9.999971 

.999969 

.999968 

.999966 

.999964 

.999963 

.999961 

.999959 

.999958 

.999956 

9.999954 

.999952 

.999950 

.999948 

.999946 

.999944 

.999942 

.999940 

.99993S 

.999936 

.999934 

.00 

.00 

.00 

.00 

.00 

.00 

.00 

.00 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.04 

.04 

.04 

Inf. neg. 

6.463726 
.764756 
.940847 

7.065786 

.162696 

.241878 

.308825 

.366817 

.417970 

7.463727 
.505120 
.542909 
.577672 
.609857 
.639820 
.667849 
.694179 
.719003 
.742484 

7.764761 

.785951 

.806155 

.S25460 

.843944 

.S61674 

.878708 

.895099 

.9I0S94 

.926134 

7.940858 

.955100 

.968889 

.982253 

.995219 

8.007809 

.020044 

.031945 

.043527 

.054809 

8.0658C6 
.07653 ; 
.086997 
.097217 
.107203 
.116963 
.126510 
.135851 
.144996 
.153952 

8.162727 

.171328 

.179763 

.188036 

.196156 

.204125 

.211953 

.219641 

.227195 

.234621 

.241921 

5017.17 
2934.85 
2082.31 

1615.17 
1319.69 
1115.78 

966.54 

852.55 

762.63 

689.88 

629.81 
579.37 

536.42 

499.39 

467.15 

438.82 

413.73 
391.36 
371.28 

353.16 

336.73 
321.76 

■ 303.07 
295.49 

283.90 
273.18 

263.25 
254.01 

245.40 

237.35 
'229.82 

222.75 
216.10 

209.83 
203.92 

198.33 
193.05 
188.03 

183.27 

178.75 

174.44 

170.34 

166.42 
162.68 
159.11 
155.69 

152.41 

149.27 

146.25 

143.36 
140.57 

137.90 
135.32 

132.84 

130.44 
128.14 

125.91 

123.76 

121.63 

Infinite. 

3.536274 

.235244 

.059153 

2.934214 

.837304 

.758122 

.691175 

.633183 

.582030 

2.536273 

.494880 

.457091 

.422328 

.390143 

.360180 

.332151 

.305821 

.280997 

.257516 

2.235239 

.214049 

.193845 

.174540 

.156056 

.138326 

.121292 

.104901 

.089106 

.073866 

2.059142 

.044900 

.031111 

.017747 

.004781 

1.992191 
.979956 
.96SO55 
.956473 
.945191 

1.934194 

.923469 

.913003 

.902783 

.892797 

.883037 

.873490 

.864149 

.855004 

.846048 

1.837273 

.828672 

.820237 

.811964 

.803844 

.795874 

.788047 

.780359 

.772805 

.765379 

.758079 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


90° 


89 





































COSINES, TANGENTS, AND COTANGENTS. 175 

1 ° 178a 


M. 

Sine 

D. V-. 

Cosine. 

D r. 

Tang. 

D. 1". 

1 

1 Cotangc 

M. 

0 

1 

2 

3 

4 

5 

6 
/ 

8 

9 

tO 

11 

12 

13 

14 
1.5 
.8 

17 

18 

19 

20 
21 
22 
23 
2-1 
25 
28 

27 

28 

29 

30 

31 

32 

33 
•G 
35 

38 
:.G 
33 

39 

40 
11 

42 

43 

41 
45 
18 
47 
43 

; 49 

50 

tn 

•52 

53 

54 

55 

56 

57 
53 

59 

60 

8.241855 

.249033 

.256094 

.263042 

.269831 

.276614 

.283243 

.239773 

.296207 

.302546 

8.30S794 

.314954 

.321027 

.327016 

.332924 

.338753 

.314504 

.350181 

.355783 

.361315 

8.366777 

.372171 

.377499 

.332762 

.387962 

.393101 

.398179 

.403199 

.403161 

,.413063 

S.417919 

.422717 

.427462 

.432156 

.436300 

.441394 

.445941 

.450440 

.454893 

.459301 

8.463665 

.467935 

.472263 

.476493 

.480693 

.484848 

.488963 

.493040 

.497078 

.501030 

3.505045 
.503974 
.512867 
.516726 
.520551 
.524343 
.528102 
.531823 
.535523 
.539186 
.542819 

119.63 

117.69 

115.80 
' 113.93 

112.21 

110.50 

108.83 

107.22 

105.66 
104.13 

102.66 

101.22 
99.82 

98.47 
97.14 

95.86 

94.60 
93.33 

92.19 
91.03 

89.90 

88.80 

87.72 
86.67 

85.64 

84.64 
83.66 
82.71 

81.77 

80.86 

79.96 

79.09 

78.23 
77.40 

76.58 

75.77 
74.99 
74.22 

73.47 

72.73 

72.00 

71.29 

70.60 

69.91 

69.24 

68.59 
67.94 

67.31 

66.69 
66.08 

65.48 
64.89 

64.32 
63.75 

63.19 

62.65 
62.11 
61.53 
61.06 
60.55 

9.999934 

.999932 

.999929 

.999927 

.999925 

.999922 

.999920 

.999913 

.899915 

.999913 

9.999910 

.999907 

.999905 

.999902 

.999899 

.999397 

.999394 

.999891 

.999888 

.999885 

9.999882 

.999379 

.999876 

.999873 

.999370 

.999367 

.999864 

.999361 

.999858 

.999854 

9.999851 

.999348 

.999844 

.999841 

.999833 

.999834 

.999831 

.999827 

.999324 

.999820 

9.999316 

.999813 

.999809 

.999805 

.999301 

.999797 

.999794 

.999790 

.999786 

.999782 

9.999778 

.999774 

.999769 

.999765 

.999761 

.999757 

.999753 

.999748 

.999744 

.999740 

.999735 

* .04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 

.04 

.04 

.04 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

8.241921 

.249102 

.256165 

.263115 

.269956 

.276691 

.233323 

.289856 

.296292 

.302634 

8.308884 

.315046 

.321122 

.327114 

.333025 

.333856 

.344610 

.350289 

.355895 

.361430 

8.366395 

.372292 

.377622 

.382889 

.388092 

.393234 

.398315 

.403338 

.408304 

.413213 

8.418068 

.422869 

.427618 

.432315 

.436962 

.441560 

.446110 

.450613 

.455070 

.459481 

8.463S49 

.468172 

.472454 

.476693 

.480892 

.485050 

.489(70 

.493250 

.497293 

.501298 

8.505267 

.509200 

.513098 

.516961 

.520790 

.524586 

.528349 

.532080 

.535779 

.539447 

.543084 

119.67 

117.72 

115.84 

114.02 

112.25 
110.54 
108.87 

107.26 

105.70 

104.18 

102.70 

101.26 
99.87 
98.51 
97.19 

95.90 

94.65 
93.43 
92.24 
91.08 

89.95 
88.85 
87.77 
86.72 

85.70 
84.69 

83.71 

82.76 
81.82 

80.91 

80.02 

79.14 
78.29 
77.45 
76.63 
75.83 
75.05 
74.23 
73.53 
72.79 

72.06 

71.35 

70.66 
69.98 
69.31 
68.65 
68.01 

67.38 

66.76 

66.15 

65.55 

64.96 

64.39 
63.82 
63.26 

62.72 

62.18 
01.65 
61.13 
60.62 

1.758079 

.750898 

.743335 

.736885 

.730044 

.723309 

.716677 

.710144 

.703708 

.697366 

1.691116 

.684954 
.678878 
.672886 
.666975 
.661144 
.655390 
.649711 
.644105 
.638570 

1.633105 

.627708 

.622378 

.617111 

.611908 

.606766 

.601685 

.596662 

.591696 

.586787 

1.581932 

.577131 

.572332 

.567635 

.563038 

.558440 

.553890 

.549387 

.544930 

.540519 

1.536151 

.531828 

.527546 

.523307 

.519103 

.514950 

.510330 

.506750 

.502707 

.498702 

1.494733 

.490800 

.486902 

.483039 

.479210 

.475414 

.471651 

.467920 

.464221 

.460553 

.456916 

-60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

L 

Cosine. 1 

D. 1". 

Sine. 

D. 1". 

Cotang. 1 

D. 1". 

Tang. 

M. 


91 ° 


880 























































176 


TABLE XIII. LOGARITHMIC SINES, 

171 * 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 
27 
23 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

8.542819 

.546422 

.549995 

.553539 

.557054 

.560540 

.563999 

.567431 

.570836 

.574214 

8.577566 

.580392 

.584193 

.587469 

.590721 

.593948 

.597152 

.600332 

.603439 

.606623 

8.609734 
.612823 
.615891 
.618937 
.621962 
.624965 
.627948 
.630911 
.633854 
.636776 

8.639680 

.642563 

.645423 

.648274 

.651102 

.653911 

.656702 

.659475 

.662230 

.664968 

8.667689 

.670393 

.673080 

.675751 

.678405 

.631043 

.633665 

.636272 

.688863 

.691433 

8.693993 

.696543 

.699073 

.701589 

.704090 

.706577 

.709049 

.711507 

.713952 

.716383 

.718800 

60.04 

59.55 

59.06 

58.58 

58.11 
57.65 

57.19 

56.74 
56.30 
55.87 

55.44 

55.02 

54.60 

54.19 

53.79 
53.39 
53.00 

52.61 

52.23 
51.86 

51.49 

51.12 
50.77 

50.41 
50.06 
49.72 

49.38 
49.04 
48.71 

48.39 

4S.06 

47.75 

47.43 

47.12 
46.82 
46.52 
46.22 
45.93 

45.63 
45.35 

45.07 

44.79 

44.51 

44.24 

43.97 
43.70 

43.44 
43.18 

42.92 
42.67 

42.42 
42.17 

41.93 

41.63 

41.44 
41.21 

40.97 
40.74 

40.51 
40.29 

9.999735 

.999731 

.999726 

.999722 

.999717 

.999713 

.999708 

.999704 

.999699 

.999694 

9.999689 

.999685 

.999680 

.999675 

.999670 

.999665 

.999660 

.999655 

.999650 

.999645 

9.999640 

.999635 

.999629 

.999624 

.999619 

.999614 

.999608 

.999603 

.999597 

.999592 

9.999586 

.999581 

.999575 

-.999570 

.999564 

.999558 

.999553 

.999547 

.999541 

.999535 

9.999529 

.999524 

.999518 

.999512 

.999506 

.999500 

.999493' 

.999487 

.999481 

.999475 

9.999469 

.999463 

.999456 

.999450 

.999443 

.999437 

.999431 

.999424 

.999418 

.999411 

.999404 

■%r 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.03 

.08 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

8.543084 

.546691 

.550268 

.553817 

.557336 

.560828 

.564291 

.567727 

.571137 

.574520 

8.577877 

.531208 

.584514 

.587795 

.591051 

.594283 

.597492 

.600677 

.603839 

.606978 

8.610094 

.613189 

.616262 

.619313 

.622343 

.625352 

.623340 

.631308 

.634256 

.637184 

8.640093 

.642982 

.645853 

.648704 

.651537 

.654352 

.657149 

.659928 

.662689 

.665433 

8.663160 

.670870 

.673563 

.676239 

.678900 

.681544 

.6S4172 

.6S6784 

.639381 

.691963 

8.694529 

.697081 

.699617 

.702139 

.704646 

.707140 

.709618 

.712083 

.714534 

.716972 

.719396 

60.12 

59.62 

59.14 
58.66 
58.19 

57.73 

57.27 
56.82 
56.38 
55.95 

55.52 
55.10 
54.68 

54.27 

53.87 

53.47 
53.08 
52.70 
52.32 
51.94 

51.58 

51.21 
50.85 
50.50 

50.15 
49.81 

49.47 
49.13 

48.80 

48.48 

48.16 

47.84 

47.53 

47.22 
46.91 

46.61 

46.31 
46.02 

45.73 
45.45 

45.16 

44.88 

44.61 
44.34 
44.07 

43.80 

43.54 

43.28 
43.03 
42.77 

42.52 

42.23 
42.03 
41.79 

41.55 

41.32 
41.03 

40.85 

40.62 
40.40 

1.456916 

.453309 

.449732 

.446183 

.442664 

.439172 

.435709 

.432273 

.428863 

.425480 

1.422123 

.418792 

.415486 

.412205 

.408949 

.405717 

.402508 

.399323 

.396161 

.393022 

1.389906 

.386811 

.383738 

380687 

377657 

.374648 

.371660 

.368692 

.365744 

.362816 

1.359907 
.357018 
.354147 
.351296 
.348463 
.345648 
.342851 
.340072 
.337311 
.334567 

1.331840 

.329130 

.326437 

•323761 

.321100 

.318456 

.315828 

.313216 

.310619 

.308037 

1.305471 

.302919 

.300383 

.297861 

.295354 

.292860 

.290382 

.287917 

.285466 

.283028 

.280604 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 




S7- 





































So 


COSINES, TANGENTS, AND COTANGENTS. 177 

178° 


M 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 
31 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

8.718800 

.721204 

.723595 

.725972 

.728337 

.730688 

.733027 

.735354 

.737667 

.739969 

8.742259 

.744536 

.746802 

.749055 

.751297 

.753523 

.755747 

.757955 

.760151 

.762337 

8.761511 

.766675 

.763823 

.770970 

.773101 

.775223 

.777333 

.779434 

.781524 

.783605 

8.785675 

.787736 

.789787 

.791823 

.793359 

.795881 

.797894 

.799397 

.801892 

.803376 

8.805852 

.807819 

.809777 

.811726 

.813667 

.815599 

.817522 

.819436 

.821343 

.823240 

8.825130 

.827011 

.823884 

.830749 

.832607 

.834456 

.836297 

.833130 

.839956 

.841774 

.813585 

40.06 

39.84 

39.62 

39.41 

39.19 
38.98 

38.77 
38.57 

38.36 

38.16 

37.96 

37.76 
37.56 

37.37 

37.17 

36.93 

36.80 
36.61 

36.42 
36.24 

36.06 

35.88 

35.70 

35.53 

35.35 

35.18 
35.01 

31.81 
34.67 
34.51 

34.31 

34.18 
34.02 
33.86 

33.70 

33.54 
33.39 
33.23 
33.08 

32.93 

32.78 

32.63 

32.49 
32.34 

32.20 
32.05 
31.91 

31.77 

31.63 

31.49 

31.36 
31.22 
31.08 
30.95 

30.82 
30.69 
30 56 

30.43 
30.30 
30.17 

9.999404 

.999398 

.999391 

.999384 

.999378 

.999371 

.999361 

.999357 

.999350 

.999343 

9.999336 

.999329 

.999322 

.999315 

.999308 

.999301 

.999294 

.999287 

.999279 

.999272 

9.999265 

.999257 

.999250 

.999242 

.999235 

.999227 

.999220 

.999212 

.999205 

.999197 

9.999189 

.999181 

.999174 

.999166 

.999158 

.999150 

.999142 

.999131 

.999126 

.999118 

9.999110 

.999102 

.999094 

.999086 

.999077 

.999069 

.999061 

.999053 

.999044 

.999036 

9.999027 

.999019 

.999010 

.999002 

.998993 

.998931 

.993976 

.998967 

.998958 

.998950 

.998941 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.15 

.15 

.15 

.15 

8.719396 

.721806 

.724204 

.726588 

.728959 

.731317 

.733663 

.735996 

.738317 

.740626 

8.742922 

.745207 

.747479 

.749740 

.751989 

.754227 

.756453 

.758668 

.760872 

.763065 

8.765246 

.767417 

.769578 

.771727 

.773866 

.775995 

.778114 

.780222 

.782320 

.784408 

8.786486 

.788554 

.790613 

.792662 

.794701 

.796731 

.793752 

.800763 

.802765 

.804758 

8.806742 

.808717 

.810683 

.812641 

.814589 

.816529 

.818461 

.820384 

.822298 

.824205 

8.826103 

.827992 

.829374 

.831748 

.833613 

.835471 

.837321 

.839163 

.840998 

.842825 

.844644 

40.17 

39.95 
39.74 

39.52 

39.31 

39.10 
38.89 

38.68 

38.48 
33.27 

38.07 

37.88 

37.68 

37.49 
37.29 

37.10 

36.92 
36.73 
36.55 

36.36 

36.18 
36.00 

35.83 
35.65 
35.48 

35.31 

35.14 
34.97 
34.80 
34.64 

34.47 

34.31 

34.15 
33.99 

33.83 
33.63 

33.52 

33.37 

33.22 
33.07 

32.92 

32.77 

32.62 

32.48 
32.33 

32.19 
32.05 
31.91 

31.77 

31.63 

31.50 
31.36 

31.23 
31.09 

30.96 

30.83 
30.70 
30.57 
30.45 

30.32 

1.280604 

.273194 

.275796 

.273412 

.271041 

.268683 

.266337 

.264004 

.261633 

.259374 

1.25707S 
.254793 
.252521 
.250260 
.248011 
.245773 
.243547 
.241332 
.239123 
.236935 

1.234754 

.232583 

.230422 

.228273 

.226134 

.224005 

.221886 

.219778 

.217680 

.215592 

1.213514 

.211446 

.209387 

.207338 

.205299 

.203269 

.201248 

.199237 

.197235 

.195242 

1.193259 

.191283 

.189317 

.187359 

.185411 

.183471 

.181539 

.179616 

.177702 

.175795 

1.173397 

.172003 

.170126 

.168252 

.166387 

.164529 

.162679 

.160837 

.159002 

.157175 

.155356 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
* 11 

10 

9 

a 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. I 

D. 1". 

Sine. 

D. 1". 

Cofcang. 

D. 1". 

Tang. 

M. 


i3 3 9 88° 

































178 TABLE XIII. LOGARITHMIC SINES, 

40 1759 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D 1". 

1 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 
17 
13 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 
67 
53 

59 

60 

8.843585 

.845387 

.847183 

.848971 

.850751 

.852525 

.854291 

.856049 

.857801 

.859546 

8.861283 

.863014 

.864738 

.866455 

.868165 

.869868 

.871565 

.873255 

.874938 

.876615 

8.878285 

.879949 

.881607 

.883258 

.884903 

.886542 

.888174 

.889801 

.891421 

.893035 

8.894643 

.896246 

.897842 

.899432 

.901017 

.902596 

.904169 

.905736 

.907297 

.908853 

8.910404 

.911949 

.913488 

.915022 

.916550 

.918073 

.919591 

.921103 

.922610 

.924112 

8.925609 

.927100 

.928587 

.930063 

.931544 

.933015 

.934431 

.935942 

.937398 

.933850 

.940296 

30.05 

29.92 

29.80 

29.68 

29.55 
29.43 

29.31 

29.19 
29.08 
28.96 

28.84 

28.73 
28.61 

28.50 
28.39 
28.28 
28.17 
28.06 
27.95 

27.84 

27.73 
27.63 
27.52 

27.42 

27.31 

27.21 

27.11 
27.00 
26.90 

26.80 

26.70 

26.60 

26.51 
26.41 

26.31 

26.22 

26.12 
26.03 

25.93 

25.84 

25.75 

25.66 

25.56 
25:47 
25.38 
25.29 
25.21 
25.12 
25.03 

24.94 

24.86 

24.77 

24.69 
24.60 

24.52 

24.43 
24.35 
24.27 

24.19 
24.11 

9.998941 

.998932 

.998923 

.998914 

.998905 

.998896 

.99SS87 

.998878 

.998869 

.998860 

9.998851 

.998841 

.998832 

.998823 

.998813 

.998804 

.998795 

.993785 

.998776 

.99S766 

9.99S757 

.998747 

.998738 

.998728 

.998718 

.998708 

.998699 

.998689 

.998679 

.998669 

9.998659 

.998649 

.998639 

.998629 

.998619 

.998609 

.998599 

.998589 

.998578 

.998568 

9.998558 

.998548 

.998537 

.998527 

.998516 

.998506 

.998495 

.998485 

.998474 

.998464 

9.998453 

.998442 

.998431 

.998421 

.998410 

.998399 

.998388 

.998377 

.998366 

.998355 

.998344 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.18 

.18 

.18 

.18 

.18 

.18 

18 

.18 

18 

.18 

.18 

.18 

.18 

.18 

.18 

8.844644 

.846455 

.848260 

.850057 

.851846 

.853628 

.855403 

.857171 

.858932 

.860686 

8.862433 

.864173 

.865906 

.867632 

.869351 

.871064 

.872770 

.874469 

.876162 

.877849 

8.879529 

.881202 

.882869 

.884530 

.886185 

.887833 

.889476 

.891112 

.892742 

.894366 

8.895984 

.897596 

.899203 

.900803 

.902398 

.903987 

.905570 

.907147 

.908719 

.910285 

8.911846 

.913401 

.914951 

.916495 

.918034 

.919568 

.921096 

.922619 

.924136 

.925649 

8.927156 

.928658 

.930155 

.931647 

.933134 

.934616 

.936093 

.937565 

.939032 

.940494 

.941952 

30.20 
30.07 

29.95 

29.83 

29.70 

29.58 

29.46 
29.35 
29.23 

29.11 

29.00 

28.88 

28.77 
28.66 

28.55 
28.43 
28.32 
28.22 

28.11 
28.00 

27.89 

27.79 

27.68 

27.58 

27.47 

27.37 
27.27 
27.17 
27.07 
26.97 

26.87 

26.77 
26.67 

26.58 

26.48 
26.39 

26.29 

26.20 
26.10 
26.01 

25.92 

25.83 
25.74 
25.65 

25.56 
25.47 

25.38 
. 25.29 

25.21 

25.12 

25.04 

24.95 

24.87 

24.78 

24.70 
24.62 
24.53 
24.45 
24.37 

24.29 

1.155356 

.153545 

.151740 

.149943 

.148154 

.146372 

.144597 

.142829 

.141068 

.139314 

1.137567 

.135827 

.134094 

.132368 

.130649 

.128936 

.127230 

.125531 

.123838 

.122151 

1.120471 

.118798 

.117131 

.115470 

.113815 

.112167 

.110524 

.108888 

.107258 

.105634 

1.104016 

.102404 

.100797 

.099197 

.097602 

.096013 

.094430 

.092853 

.091281 

.089715 

1.088154 

.086599 

.085049 

.083505 

.081966 

.080432 

.078904 

.077381 

.075864 

.074351 

1.072844 

.071342 

.069845 

.068353 

.066866 

.065384 

.063907 

.062435 

.060968 

.059506 

.058048 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1'. 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. | 

M 


853 ■ 


94.0 







































COSINES, TANGENTS, AND COTANGENTS. 179 

5° 174:0 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1'. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 
17 
IS 

19 

20 
21 
22 
23 
21 

25 

26 
27 
23 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

8.940296 

.941733 

.943174 

.944606 

.916034 

.917456 

.943874 

.950237 

.951696 

.953100 

8.951499 

.955394 

.957234 

.958670 

.960052 

.961429 

.962301 

.964170 

.965534 

.966393 

8.963249 

.989600 

.970947 

.972289 

.973623 

.974902 

.976293 

.977619 

.973941 

.930259 

8.931573 

.932333 

.934189 

.935491 

.936789 

.933033 

.939374 

.990660 

.991943 

.993222 

8.994497 

.995763 

.997036 

.998299 

.999560 

9.000316 

.002069 

.003318 

.004563 

.005305 

9.007044 

.003278 

.009510 

.010737 

.011962 

.013182 

.014400 

.015613 

.016324 

.018031 

.019235 

24.03 

23.95 

23.87 
23.79 
23.71 

23.63 
23.55 
23.48 

23.40 

23.32 

23.25 

23.17 

23.10 
23.02 

22.95 

22.88 
22.81 
22.73 
22.66 
22.59 

22.52 

22.45 

22.33 

22.31 

22.24 

22.17 

22.10 
22.03 
21.97 
21.90 

21.83 
21.77 

21.70 

21.64 
21.57 

21.51 
21.44 

21.33 

21.31 

21.25 

21.19 
21.12 
21.06 
21.00 
■ 20.94 

20.83 
20.82 
20.76 

20.70 

20.64 

20.53 

20.52 

20.46 

20.40 
20.35 
20.29 
20.23 

20.17 
20.12 
20.06 

9.993344 

.998333 

.993322 

.998311 

.993300 

.993239 

.993277 

.993266 

.993255 

.993243 

9.993232 

.993220 

.993209 

.993197 

.993136 

.993174 

.993163 

.998151 

.993139 

.998123 

9.998116 

.993104 

.993092 

.993030 

.993068 

.993056 

.993044 

.993032 

.998020 

.998008 

9.997996 

.997984 

.997972 

.997959 

.997917 

.997935 

.997922 

.997910 

.997897 

.997385 

9.997872 

.997360 

.997847 

.997835 

.997822 

.997809 

.997797 

.997734 

.997771 

.997753 

9.997745 

.997732 

.997719 

.997706 

.997693 

.997630 

.997667 

.997654 

.997641 

.997623 

.997614 

.18 

.19 

.19 

.19 

.19 

.19 

.19 

.19 

.19- 

.19 

.19 

.19 

.19 

.19 

.19 

.19 

.19 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.21 

.21 

.21 

.21 

.21 

.21 

21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.22 

.22 

.22 

.22 

.22 

.22 

.22 

.22 

.22 

.22 

8.941952 

.943404 

.944852 

.946295 

.947734 

.949168 

.950597 

.952021 

.953441 

.954356 

8.956267 

.957674 

.959075 

.960473 

.961866 

.963255 

.964639 

.966019 

.967394 

.963766 

8.970133 

.971496 

.972355 

.974209 

.975560 

.976906 

.978248 

.979586 

.9S092I 

.932251 

8.933577 

.934S99 

.936217 

.9S7532 

.938342 

.990149 

.991451 

.992750 

.994045 

.995337 

8.996624 

.997903 

.999188 

9.000465 

.001733 

.003007 

.004272 

.005534 

.006792 

.003047 

9.009293 

.010546 

.011790 

.013031 

.014263 

.015502 

.016732 

.017959 

.019183 

.020403 

.021620 

24.21 

24.13 
24.05 

23.97 

23.90 
23.82 

23.74 
23.67 
23.59 

23.51 

23.44 

23.36 

23.29 

23.22 

23.14 
23.07 
23.00 
22.93 
22.86 

22.79 

22.72 

22.65 

22.58 

22.51 

22.44 

22.37 

22.30 
22.24 
22.17 
22.10 

22.04 

21.97 

21.91 

21.84 
21.78 
21.71 

21.65 

21.59 

21.52 
21.46 

21.40 

21.34 

21.27 
21.21 

21.15 
21.09 
21.03 

20.97 

20.91 

20.85 

20.80 

20.74 
20.63 
20.62 
20.56 
20.51 

20.45 
20.39 

20.34 

20.28 

1.058048 

.056596 

.055148 

.053705 

.052266 

.050832 

.049403 

.047979 

.046559 

.045144 

1.043733 

.042326 

.040925 

.039527 

.038134 

.036745 

.035361 

.033981 

.032606 

.031234 

1.029367 
- .023504 
.027145 
.025791 
.024440 
.023094 
.021752 
.020414 
.019079 
.017749 

1.046423 

.015101 

.013783 

.012468 

.011158 

.009851 

.008549 

.007250 

.005955 

.004663 

1.003376 

.002092 

.000812 

0.999535 

.993262 

.996993 

.995728 

.994466 

.993208 

.991953 

0.990702 

.939454 

.988210 

.986969 

.985732 

.984498 

.983268 

.932041 

.9S0317 

.979597 

.978330 

6C 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

o ! 

mV j 

III. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 


D5° 


840 

































180 TABLE XIII. LOGARITHMIC SINES,' 

GO 1731 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.019235 

.020435 

.021632 

.022825 

.024016 

.025203 

.026386 

.027567 

.028744 

.029918 

9.031089 

.032257 

.033421 

.034582 

.035741 

.036896 

.038048 

.039197 

.040342 

.041485 

9.042625 

.013762 

.044895 

.046026 

.047154 

.048279 

.049400 

.050519 

.051635 

.052749 

9.053359 

.054966 

.056071 

.057172 

.058271 

.059367 

.060460 

.061551 

.062639 

.063724 

9.064806 

.065885 

.066962 

.068036 

.069107 

.070176 

.071242 

.072306 

.073366 

.074424 

9.075480 

.076533 

.077583 

.078631 

.079676 

.030719 

.081759 

.082797 

.083832 

. .084864 
.0S5894 

20.00 

19.95 

19.89 

19.84 
19.78 
19.73 

19.67 
19.62 
19.57 
19.51 

19.46 

19.41 

19.36 

19.30 

19.25 

19.20 
19.15 
19.10 
19.05 
19.00 

18.95 

18.90 

18.85 
18.80 
18.75 
18.70 
18.65 
18.60 

18.55 

18.50 

18.46 

18.41 

18.36 

18.31 
18.27 
18.22 

18.17 
18.13 
18.08 
18.04 

17.99 

17.95 

17.90 

17.86 
17.81 
17.77 
17.72 

17.68 
17.64 
17.59 

17.55 

17.51 

17.46 

17.42 
17.38 
17.34 
17.29 

17.25 

17.21 

17.17 

9.997614 

.997601 

.997588 

.997574 

.997561 

.997547 

.997534 

.997520 

.997507 

.997493 

9.997480 

.997466 

.997452 

.997439 

.997425 

.997411 

.997397 

.997383 

.997369 

.997355 

9.997341 

.997327 

.997313 

.997299 

.997285 

.997271 

.997257 

.997242 

.997228 

.997214 

9.997199 

.997185 

.997170 

.997156 

.997141 

.997127 

.997112 

.997098 

.997083 

.997068 

9.997053 

.997039 

.997024 

.997009 

.996994 

.996979 

.996964 

.996949 

.996934 

.996919 

9.996904 

.996889 

.996874 

.996858 

.996843 

.996328 

.996312 

.996797 

.996782 

.996766 

.996751 

.22 

.22 

.22 

.22 

.22 

.22 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.26 

.26 

.26 

.26 

.26 

.26 

9.021620 

.022834 

.024044 

.025251 

.026455 

.027655 

.028852 

.030016 

.031237 

.032425 

9.033609 

.034791 

.035969 

.037144 

.038316 

.039485 

.040651 

.041813 

.042973 

.044130 

9.045284 

.046434 

.047582 

.048727 

.049869 

.051008 

.052144 

.053277 

.054407 

.055535 

9.056659 

.057781 

.058900 

.060016 

.061130 

.062240 

.063348 

.064453 

.065556 

.066655 

9.067752 

.068846 

.069938 

.071027 

.072113 

.073197 

.074278 

.075356 

.076432 

.077505 

9.078576 

.079644 

.080710 

.081773 

.082833 

.083891 

.084947 

.OS6000 

.087050 

.088098 

.089144 

20.23 

20.17 
20.12 
20.06 
20.01 
19.95 
19.90 
19.85 

19.79 

19.74 

19.69 

19.64 

19.58 
19.53 
19.48 
19.43 
19.38 

19.33 

19.28 

19.23 

19.18 
19.13 
19.08 
19.03 
18.98 

18.93 

18.89 

18.84 

18.79 

18.74 

18.70 

18.65 
18.60 
18.56 

18.51 

18.46 

18.42 
18.37 

18.33 

18.28 

18.24 

18.19 
18.15 
18.10 
18.06 
18.02 ‘ 
17.97 

17.93 

17.89 

17.84 

17.80 
17.76 
17.72 
17.67 
17.63 

17.59 
17.55 

17.51 

17.47 

17.43 

0.978380 

.977166 

.975956 

.974749 

.973545 

.972345 

.971148 

.969954 

.968763 

.967575 

0.966391 
.965209 
.964031 
.962856 
.9616S4 
.960515 
.959349 
.958187 
.957027 
.955870 

0.954716 

.953566 

.952418 

.951273 

.950131 

.948992 

.947856 

.946723 

.945593 

.944465 

0.943341 

.942219 

.941100 

.939984 

.938870 

.937760 

.936652 

.935547 

.934444 

.933345 

0.832248 
.931154 
.930062 
.928973 
.927887 
.926803 
.925722 
.924644 
.923568 
.922495 

0.921424 

.920356 

.919290 

.918227 

.917167 

.916109 

.915053 

.914000 

.912950 

.911902 

.S108E6 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

4G 

47 

46 

45 

14 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M 


83 « 































70 


COSINES, TANGENTS, AND COTANGENTS 


181 


17 2 r -' 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.085394 

.086922 

.037947 

.033970 

.039990 

.091008 

.092024 

.093037 

.094047 

.095056 

9.096062 

.097065 

.093066 

.099065 

.100062 

.101056 

.102048 

.103037 

.104025 

.105010 

9.105992 

.106973 

.107951 

.105927 

.109901 

.110873 

.111842 

.112309 

.113774 

.114737 

9.115693 

.116656 

.117613 

.118567 

.119519 

.120469 

.121417 

.122362 

.123306 

.124248 

9.125187 

.126125 

.127060 

.127993 

.128925 

.129354 

.130731 

.131706 

.132630 

.133551 

9.134470 

.135337 

.136303 

.137216 

.133123 

.139037 

.139944 

.140850 

.141754 

.142655 

.143555 

17.13 
17.09 
17.05 
17.00 
16.96 
16.92 
16.88 
16.84 
16.80 

16.76 

16.73 

16.69 

16.65 
16.61 
16.57 
16.53 

16.49 
16.46 

16.42 

16.33 

16.34 
16.30 
16.27 
16.23 

16.19 
16. 16 
16.12 
16.03 
16.05 
16.01 

15.98 

15.94 

15.90 

15.87 

15.83 

15.80 

15.76 

15.73 

15.69 

15.66 

15.62 

15.59 

15.56 

15.52 

15.49 
15.45 

15.42 
15.39 

15.35 
15.32 

15.29 

15.26 

15.22 

15.19 
15.16 

15.13 
15.09 
15.06 
15.03 
15.00 

9.996751 

.996735 

.996720 

.996704 

.996633 

.996673 

.996657 

.996641 

.996625 

.996610 

9.996594 

.996573 

.996562 

.996546 

.996530 

.996514 

.996498 

.996482 

.996465 

.996449 

9.996433 

.996417 

.996400 

.996334 

.996363 

.996351 

996335 

.99S318 

.996302 

.996235 

9.996269 

.996252 

.996235 

.996219 

.996202 

.996185 

.996163 

.996151 

.996134 

.996117 

9.996100 

.996033 

.996066 

.996049 

.996032 

.996015 

.995998 

.995930 

.995963 

.995946 

9.995928 
.995911 
.995894 
.995876 
.995859 . 
.995341 
.995S23 
.995806 
.995788 
.995771 
.995753 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 
.27 
.27 
.27 
- .27 
.27 
.23 
.28 
.23 
.23 

.28 

.28 

.23 

.23 

.23 

.23 

.23 

.28 

.23 

.23 

.23 

.28 

.28 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.30 

9.089144 

.090187 

.091228 

.092266 

.093302 

.094336 

.095367 

.096395 

.097422 

.093446 

9.099468 

.100487 

•101504 

.102519 

.103532 

.104542 

.105550 

.106556 

.107559 

.108560 

9.109559 
.110556 
.111551 
.112543 
.113533 
.114521 
.115507 
.116191 
.117472 
.118452 

9.119429 

.120404 

.121377 

.122318 

.123317 

.124284 

.125249 

.126211 

.127172 

.128130 

9.129037 

.130041 

.130994 

.131944 

.132393 

.133339 

.134784 

.135726 

.136667 

.137605 

9.138542 

.139476 

.140409 

.141340 

.142269 

.143196 

.144121 

.145044 

.145966 

.146885 

.147803 

17.39 

17.35 

17.31 
17.27 
17.23 
17.19 

17.15 

17.11 
17.07 
17.03 

16.99 

16.95 

16.91 

16.83 

16.84 
16.80 

16.76 
16.72 
16.69 
16.65 

16.61 

16.58 

16.54 

16.50 

16.47 
16.43 

16.39 

16.36 

16.32 

16.29 

16.25 

16.22 

16.18 

16.15 

16.11 
16.08 
16.04 
16.01 

15.93 

15.94 

15.91 
15.87 

15.84 * 
15.81 

15.77 
15.74 
15.71 

15.63 

15.64 
15.61 

15.58 

15.55 

15.51 

15.48 
15.45 
15.42 

15.39 

15.36 

15.32 

15.29 

0.910356 

.909813 

.908772 

.907734 

.906698 

.905664 

.904633 

.903605 

.902578 

.901554 

0.900532 

.899513 

.893496 

.897481 

.896463 

.895458 

.894450 

.89:3444 

.892141 

.891440 

0.890441 

.889444 

.883449 

.887457 

.886467 

.835479 

.884493 

.883509 

.832523 

.881548 

0.880571 

.879596 

.878623 

.877652 

.876633 

.875716 

.874751 

.873789 

.872828 

.871870 

0.870913 

.869959 

.869006 

.868056 

.867107 

.866161 

.865216 

.864274 

.863333 

.862395 

0.861458 

.860524 

.859591 

.858660 

.857731 

.856804 

.855879 

.854956 

.854034 

.853115 

.852197 

60 

59 

53 
57 
'56 
55 

54 
53 
52 
51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

33 
37 
36 
35 

34 
33 
32 
31 

30 

29 

23 
27 
26 
25 

24 
23 
22 
21 

20 

19 

15 
17 

16 
15 
14 
13 
12 
11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1'. 

Tang. 

M. 


07° 


833 




































182 

8 ° 


TABLE XIII. LOGARITHMIC SINES, 

1713 


M. 

Sine 

D. 1". 

Cosine. 

D. 1' 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4- 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.143555 

.144453 

.145349 

.146243 

.147136 

.148026 

.143915 

.149S02 

.150636 

.151569 

9.152451 

.153330 

.154208 

.155083 

.155957 

.156330 

.157700 

.153569 

.159435 

.160301 

9.161164 

.162025 

.162385 

.163743 

.164600 

.165454 

.166307 

.167159 

168008 

.163856 

9.169702 

.170547 

.171389 

.172230 

.173070 

.173903 

.174744 

.175578 

.176411 

.177242 

9.178072 

.178900 

.179726 

.180551 

.181374 

.182196 

.183016 

.183834 

.184651 

.185466 

9.186230 

.187092 

.187903 

.188712 

.189519 

.190325 

.191130 

.191933 

.192734 

.193534 

.194332 

14.97 

14.93 

14.90 

14.87 

14.84 
14.81 
14.78 

14.75 

14.72 

14.69 

14.66 

14.63 

14.60 
14.57 
14.54 

14.51 

14.48 

14.45 

14.42 
14.39 

14.36 

14.33 

14.30 
14.27 
14.24 
14.22 
14.19 
14.16 
14.13 
14.10 

14.07 

14.05 

14.02 

13.99 

13.96 

13.94 

13.91 

13.88 

13.85 
13.83 

13.80 

13.77 

13.75 

13.72 

13.69 

13.67 

13.64 

13.61 
13.59 
13.56 

13 54 

13.51 

13.48 

13.46 

13.43 
13.41 

13.33 

13.36 
13.33 

13.31 

9.995753 

.995735 

.995717 

.995699 

.995631 

.995664 

.995646 

.995628 

.995610 

.995591 

9.995573 

.995555 

.995537 

.995519 

.995501 

.995482 

.995464 

.995446 

.995427 

.995409 

9.995390 

.995372 

.995353 

.995334 

.995316 

.995297 

.995278 

.995260 

.995241 

.995222 

9.995203 

.995184 

.995165 

.995146 

.995127 

.995108 

.995039 

.995070 

.995051 

.995032 

9.995013 

.994993 

.994974 

.994955 

.994935 

.994916 

.994896 

.994877 

.994857 

.994838 

9.994818 

.994798 

.994779 

.994759 

.994739 

.994720 

994700 

.994630 

.994660 

.994640 

.994620 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

9.147803 

.143718 

.149632 

.150514 

.151454 

.152363 

.153269 

.154174 

.155077 

.155978 

9.156877 

.157775 

.158671 

.159565 

.160457 

.161347 

.162236 

.163123 

.164003 

.164892 

9.165774 

.166654 

.167532 

.163409 

.169284 

.170157 

.171029 

.171899 

.172767 

.173634 

9.174499 

.175362 

.176224 

.177034 

.177942 

.178799 

.179655 

.180508 

.181360 

.182211 

9.183059 

.183907 

.184752 

.185597 

.186439 

.187280 

.188120 

.188958 

.189794 

.190629 

9.191462 

.192294 

.193124 

.193953 

.194780 

.195606 

.196430 

.197253 

.198074 

.198894 

199713 

15.26 

15.23 

15.20 

15.17 

15.14 

15.11 
15.08 
15.05 
15.02 

14.99 

14.96 

14.93 

14.90 
14.87 

14.84 

14.81 

14.78 

14.75 

14.73 

14.70 

14.67 

14.64 
14.61 
14.58 
14.56 
14.53 
14.50 
14.47 
14.44 
14.42 

14.39 

14.36 

14.33 

14.31 

14.28 

14.25 

14.23 

14.20 

14.17 

14.15 

14.12 
14.09 
14.07 
14.04 
14.02 

13.99 

13.97 

13.94 

13.91 
13.89 

13.86 

13.84 

13.81 

13.79 

13.76 

13.74 

13.71 
13.69 
13.66 

13.64 

0.852197 

.851282 

.850368 

.849456 

.848546 

.847637 

.846731 

.845826 

.844923 

.844022 

0.S43123 

.842225 

.841329 

.840435 

.839543 

.838653 

.837764 

.836877 

.835992 

.835108 

0.834226 

.833346 

.832468 

.831591 

.830716 

.829843 

.828971 

.828101 

.827233 

.826366 

0.825501 

.824638 

.823776 

.822916 

.822058 

.821201 

.820345 

.819492 

.818640 

.817789 

0.816941 

.816093 

.815248 

.814403 

.813561 

.812720 

.811880 

.811042 

.810206 

-.809371 

0.808538 

.807706 

.806876 

.806047 

.805220 

.804394 

.803570 

.802747 

.801926 

.801106 

.800287 

60 

59 

53 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Coiang. 

D. 1". 

Tang. 

M. 


eso 


810 


































•p- 


183 

170 c 


COSINES, TANGENTS, AND COTANGENTS. 


M 

Sine. 

D 1". 

Cosine. 

D. 1". 

Tang. 

D. 1”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 
45 
16 

17 

18 

19 

20 
21 
22 
23 
21 

25 

26 
27 
23 

29 

30 

31 

3 A 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.194332 

.195129 

.195925 

.196719 

.197511 

.193302 

.199091 

.199379 

.200666 

.201451 

9.202234 

.203017 

.203797 

.204577 

.205354 

.205131 

.206906 

.207679 

.203452 

.209222 

9.209992 

.210760 

.211526 

.212291 

.213055 

.213318 

.214579 

.215333 

.216097 

.216354 

9.217609 

.218363 

.219116 

.219363 

.220818 

.221367 

.222115 

.222361 

.223606 

.221349 

9.225092 

.225333 

.226573 

.227311 

.223043 

.223784 

.229518 

.230252 

.230934 

.231715 

9.232444 
.233172 
.233399 
.234625 
.235349 
'.236073 
.236795 
.237515 
.233215 
.233953 
.2396’ 1 

13.28 

13.26 

13.23 

13.21 

13.18 

13.16 
13.13 
13.11 
13.08 
13.06 

13.04 

13.01 

12.99 

12.96 

12.91 

12.92 
12.89 
12.87 
12.85 
12.82 

12.80 

12.78 

12.75 

12.73 

12.71 

12.63 
12.66 

12.64 
12.62 
12.59 

12.57 

12.55 

12.53 

12.50 

12.48 

12.46 

12.44 

12.42 

12.39 

12.37 

12.35 

12.33 

12.31 

12.29 

12.26 

12.24 

12.22 
12.20 

12.18 

12.16 

12.14 

12.12 

12.10 

12.07 

12.05 

12.03 

12.01 

11.99 

11.97 
11.95 

9.994620 

.994600 

.994580 

.994560 

.994540 

.994519 

.994499 

.994479 

.994459 

.994438 

9.994418 

.994398 

.994377 

.994357 

.994336 

.994316 

.994295 

.994274 

.994254 

.994233 

9.994212 

.994191 

.994171 

.994150 

.994129 

.994103 

.994087 

.994066 

.994045 

.994024 

9.994003 

.993982 

.993960 

.993939 

.993918 

.993397 

.993875 

.993354 

.993332 

.993311 

9.993789 

.993763 

.993746 

.993725 

.993703 

.993631 

.993660 

.993633 

.993616 

.993594 

9.993572 

.993550 

.993528 

.993506 

.993484 

.993462 

.993440 

.993418 

.993396 

.993374 

.993351 

.33 

.33 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.35 

.3-5 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.37 

.37 

.37 

.37 

.37 

.37 

.37 

.37 

.37 

.37 

9.199713 

.200529 

.201345 

.202159 

.202971 

.203782 

.204592 

.205400 

.206207 

.207013 

9.207817 

.203619 

.209420 

.210220 

.211018 

.211815 

.212611 

.213405 

.214193 

.214989 

9.215780 

.216568 

.217356 

.218142 

.218926 

.219710 

.220492 

.221272 

.222052 

.222330 

9.223607 

.224332 

.225156 

.225929 

.226700 

.227471 

.223239 

.229007 

.229773 

.230539 

9.231302 

.232065 

.232826 

.233586 

.234345 

.235103 

.235359 

.236614 

.237363 

.233120 

9.238872 

.239622 

.240371 

.241118 

.241865 

.242610 

.243354 

.244097 

.244839 

.245579 

.216319 

13.62 

13.59 

13.57 

13.54 

13.52 

13.49 

13.47 

13.45 

13.42 

13.40 

13.38 

13.35 

13.33 

13.31 

13.28 

13.26 

13.24 

13.21 

13.19 

13.17 

13.15 

13.12 

13.10 

13.03 

13.06 

13.03 

13.01 

12.99 

12.97 

12.95 

12.92 

12.90 

12.83 
12.86 

12.84 
12.82 
12.79 
12.77 
12.75 
12.73 

12.71 

12.69 

12.67 

12,65 

12.63 

12.60 

12.58 

12.56 

12.54 

12.52 

12.50 

12.48 

12.46 

12.44 

12.42 

12.40 

12.33 
12.36 

12.34 
12.32 

0.800237 

.799471 

.798655 

.797841 

.797029 

.796218 

.795403 

.794600 

.793793 

.792987 

0.792183 
.791331 
.790530 
.789780 
.788982 
■.788185 
.787389 
.786595 
.785802 
.785011 

0.784220 

.783432 

.782644 

.781858 

.781074 

.780290 

.779503 

.778728 

.777948 

.777170 

0.776393 

.775618 

.774844 

.774071 

.773300 

.772529 

.771761 

.770993 

.770227 

.769461 

0.76S698 

.767935 

.767174 

.766414 

.765655 

.764897 

.764141 

.763386 

.762632 

.761880 

0.761128 

.760378 

.759629 

.758382 

.758135 

.757390 

.756646 

.755903 

.755161 

.754421 

.753681 

60 

59 

58 

57 

56 

55 

.54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

33 
37 
36 
35 

34 
33 
32 
31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. I 

D. 1", 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 













































184 TABLE XIII. LOGARITHMIC SINES, 

103 1694 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.239670 

.240386 

.241101 

.241814 

.242526 

.243237 

.243947 

.244656 

.245363 

.246069 

9.246775 

.247478 

.248181 

.248883 

.249533 

.250282 

.250980 

.251677 

.252373 

.253067 

9.253761 

.254453 

.255144 

.255834 

.256523 

.257211 

.257898 

.258583 

.259268 

.259951 

9.260633 

.261314 

.261994 

.262673 

.263351 

.264027 

.264703 

.265377 

.266051 

.266723 

9.267395 

.268065 

.263734 

.269402 

.270069 

.270735 

.271400 

.272064 

.272726 

.273333 

9.274049 

.274708 

.275367 

.276025 

.276631 

.277337 

.277991 

.278645 

.279297 

.279948 

.230599 

11.93 

11.91 

11.89 

11.87 

11.85 

11.83 
11.81 
11.79 
11.77 
11.75 

11.73 

11.71 

11.69 

11.67 

11.65 

11.63 

11.61 

11.59 

11.58 

11.56 

11.54 

11.52 
11.50 

11.43 
11.46 

11.44 
11.42 
11.41 
11.39 
11.37 

11.35 

11.33 

11.31 

11.30 

11.23 
11.26 

11.24 
11.22 
11.20 
11.19 

11.17 

11.15 

11.13 

11.12 

11.10 

11.08 

11.06 

11.05 

11.03 

11.01 

10.99 

10.93 
10.96 

10.94 

10.92 
10.91 

10.89 

10.87 

10.86 

10.84 

9.993351 

.993329 

.993307 

.993234 

.993262 

.993240 

.993217 

.993195 

.993172 

.993149 

9.993127 

.993104 

.993081 

.993059 

.993036 

.993013 

.992990 

.992967 

.992944 

.992921 

9.992893 

.992375 

.992S52 

.992829 

.992806 

.992783 

.992759 

.992736 

.992713 

.992690 

9.992666 

.992643 

.992619 

.992596 

.992572 

.992549 

.992525 

.992501 

.992478 

.992454 

9.992430 
.992406 
.992:382 
.992.359 
.992335 
.992311 
.992287 
.992263 
.992239 
.992214 

9.992190 

.992166 

.992142 

.992118 

.992093 

.992069 

.992044 

.992020 

.991996 

.991971 

.991947 

.37 

.37 

.37 

.37 

.37 

.37 

.38 

.38 

.38 

.38 

.38 

.38 

.33 

.38 

.38 

.38 

.38 

.38 

.33 

.38 

.38 

.33 

.39 

.39 

.39 

39 

39 

39 

39 

39 

39 

39 

39 

39 

.39 

.39 

.39 

.39 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 • 

.40 

.40 

.40 

.40 

.40 

.41 

.41 

.41 

.41 

.41 

.41 

.41 

9.246319 

.247057 

.247794 

.248530 

.249264 

.249998 

.250730 

.251461 

.252191 

.252920 

9.253648 

.254374 

.255100 

.255824 

.256547 

.257269 

.257990 

.258710 

.259429 

.260146 

9.260863 

.261578 

.262292 

.263005 

.263717 

.264428 

.265138 

.265847 

.266555 

.267261 

9.267967 

.268671 

.269375 

.270077 

.270779 

.271479 

.272178 

.272876 

.273573 

.274269 

9.274964 

.275658 

.276351 

.277043 

.277734 

.278424 

.279113 

.279801 

.230488 

.281174 

9.281858 

.282542 

.283225 

.283907 

.284588 

.235268 

.285947 

.286624 

.287301 

.287977 

.288652 

12.30 
12.28 
12.26 

12.24 
12.22 
12.20 
12.18 
12.17 
12.15 
12.13 

12.11 

12.09 

12.07 

12.05 

12.03 

12.01 

12.00 

11.98 

11.96 

11.94 

11.92 

11.90 

11.89 

11.87 

11.85 

11.83 

11.81 

11.79 

11.78 

11.76 

11.74 

11.72 

11.70 

11.69 

11.67 

11.65 

11.64 

11.62 

11.60 

11.58 

11.57 

11.55 

11.53 

11.51 

11.50 

11.48 

11.46 

11.45 

11.43 

11.41 

11.40 

11.38 

11.36 

11.35 

11.33 

11.31 
11.30 
11.23 
11.26 

11.25 

0.753681 

.752943 

.752206 

.751470 

.750736 

.750002 

.749270 

.748539 

.747809 

.747080 

0.746352 

.745626 

.744900 

.744176 

.743453 

.742731 

.742010 

.741290 

.740571 

.739854 

0.739137 

.738422 

.73770S 

.736995 

.736283 

.735572 

.734862 

.734153 

.733445 

.732739 

0.732033 

.731329 

.730625 

.729923 

.729221 

.728521 

.727822 

.727124 

.726427 

.725731 

0.725036 

.724342 

.723649 

.722957 

.722266 

.721576 

.720887 

.720199 

.719512 

.718826 

0.718142 

.717458 

.716775 

,716093 

.715412 

.714732 

.714053 

.713376 

.712699 

.712023 

.711348 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 ; 
22 1 
21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


1003 















































lio 


COSINES, TANGENTS, AND COTANGENTS 


185 

168 a 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.280599 

.281248 

.281897 

.282544 

.233190 

.283336 

.284480 

.235124 

.285766 

.286408 

9.287048 

.237688 

.288326 

.233964 

.289600 

.290236 

.290370 

.291504 

.292137 

.292763 

9.293399 
.294029 
.294658 
. .295286 
.295913 
.296539 
.297164 
.297788 
.293412 
.299034 

9.299655 

.300276 

.300395 

.301514 

.302132 

.302748 

.303364 

.303979 

.304593 

.305207 

9.305819 

.306130 

.307041 

.307650 

.308259 

.308867 

.309474 

.310080 

.310635 

.311289 

9.311893 

.312495 

.313097 

.313693 

.314297 

.314897 

.315495 

.316092 

.316639 

.317234 

.317879 

10.82 

10.81 

10.79 

10.77 

10.76 

10.74 

10.72 

10.71 

10.69 

10.67 

10.66 

10.64 

10.63 

10.61 

10.59 

10.58 

10.56 

10.55 

10.53 

10.51 

10.50 

10.48 

10.47 

10.45 

10.43 

10.42 

10.40 

10.39 

10.37 

10.36 

10.34 

10.33 

10.31 

10.30 

10.28 

10.26 

10.25 

10.23 

10.22 

10.20 

10.19 

10.17 

10.16 

10.14 

10.13 

10.12 

10.10 

10.09 

10.07 

10.06 

10.04 

10.03 

10.01 

10.00 

9.93 

9.97 

9.96 

9.94 

9.93 

9.91 

9.991947 

.991922 

.991897 

.991873 

.991343 

.991823 

.991799 

.991774 

.991749 

.991724 

9.991699 

.991674 

.991619 

.991624 

.991599 

.991574 

.991549 

'.’991524 

.991498 

.991473 

9.991443 

.991422 

.991397 

.991372 

.991346 

.991321 

.991295 

.991270 

.991244 

.991218 

9.991193 

.991167 

.991141 

.991115 

.991090 

.991064 

.991038 

.991012 

.990936 

.990960 

9.990934 

.990908 

.990332 

.990355 

.990829 

.990303 

.990777 

.990750 

.990724 

.990697 

9.990671 

.990645 

.990618 

.990591 

.990565 

.990538 

.990511 

.990485 

.990458 

.990431 

.990404 

.41 

.41 

.41 

.41 

.41 

.41 

.41 

.41 

.41 

42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.45 

.45 

.45 

.45 

9.238652 

.289326 

.2S9999 

.290671 

.291342 

.292013 

.292632 

.293350 

.291017 

.294684 

9.295349 

.296013 

.296677 

.297339 

.293001 

.293662 

.299322 

.299930 

.300638 

.301295 

9.301951 

.302607 

.303261 

.303914 

.304567 

.305218 

.305869 

.306519 

.307168 

.307816 

9.303463 

.309109 

.309754 

.310399 

.311042 

.311635 

.312327 

.312963 

.313608 

.314247 

9.314885 

.315523 

.316159 

.316795 

.317430 

.318064 

.318697 

.319330 

.319961 

.320592 

9.321222 

.321851 

.322479 

.323106 

.323733 

.324358 

.324983 

.325607 

.326231 

.326853 

.327475 

11.23 

11.22 

11.20 

11.18 

11.17 

11.15 

11.14 

11.12 

11.11 

11.09 

11.07 

11.06 

11.04 

11.03 

11.01 

11.00 

10.98 

10.97 

10.95 

10.93 

10.92 

10.90 

10.89 

10.87 

10.86 

10.84 

10.83 

10.81 

10.80 

10.78 

10.77 

10.76 

10.74 

10.73 

10.71 

10.70 

10.68 

10.67 

10.65 

10.64 

10.62 

10.61 

10.60 

10.58 

10.57 

10.55 

10.54 

10.53 

10.51 

10.50 

10.48 

10.47 

10.46 

10.44 

10.43 

10.41 

10.40 

10.39 

10.37 

10.36 

0.711348 

.710674 

.710001 

.709329 

.708658 

.707987 

.707318 

.706650 

.705983 

.705316 

0.704651 

.703987 

.703323 

.702661 

.701999 

.701338 

.700678 

.700020 

.699362 

.698705 

0.693049 

.697393 

.696739 

.696086 

.695433 

.694782 

.694131 

.693481 

.692832 

.692184 

0.691537 

.690891 

.690246 

.639601 

.688958 

.638315 

.687673 

.687032 

.686392 

.685753 

0.635115 

.684477 

.683841 

.633205 

.682570 

.681936 

.681303 

.680670 

.630039 

.679408 

0.678778 

.678149 

.677521 

.676894 

.676267 

.675642 

.675017 

.674393 

.673769 

.673147 

.672525 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. I 1 '. 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


1010 


78« 








































16 ra 


186 TABLE Xlll LOGARITHMIC SINES, 


130 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 
5? 
53 

59 

60 

9.317879 

.318473 

.319066 

.319658 

.320249 

.320840 

.321430 

.322019 

.322607 

.323194 

9.323780 

.324366 

.324950 

.325534 

.326117 

.326700 

.327281 

.327862 

.328442 

.329021 

9.329599 

.330176 

.330753 

.331329 

.331903 

.332478 

.333051 

.333624 

.334195 

.334767 

9.355337 
.335906 
.336475 
.337043 
.337610 
.338176 
.338742 
.339307 
.339871 
.340434 

9.340996 

.341558 

.342119 

.342679 

.343239 

.343797 

.344355 

.344912 

.345469 

.346024 

9.346579 

.347134 

.347687 

.348240 

.348792 

.349343 

.349893 

.350443 

.350992 

.351540 

.352088 

9.90 

9.88 

9.87 

9.56 
9.84 
9.83 
9.81 
9.80 
9.79 
9.77 

9.76 
- 9.75 
9.73 
9.72 
9.70 
9.69 
9.68 
9.66 
9.65 
9.64 

9.62 

9.61 

9.60 

9.58 

9.57 
9.56 
9.54 
9.53 
9.52 
9.50 

9.49 

9.48 

9.46 

9.45 

9.44 

9.43 

9.41 

9.40 

9.39 

9.37 

9.36 

9.35 

9.34 

9.32 

9.31 

9.30 

9.29 

9.27 

9.26 

9.25 

9.24 

9.22 

9.21 

9.20 

9.19 

9.17 

9.16 

9.15 

9.14 

9.13 

9.990404' 

.990378 

.990351 

.990324 

.990297 

.990270 

.990243 

.990215 

.990188 

.990161 

9.990134 

.990107 

.990079 

.990052 

.990025 

.989997 

.989970 

.989942 

.989915 

.989887 

9.989860 

.989332 

.989804 

.989777 

.989749 

.989721 

.989693 

.989665 

.989637 

.989610 

9.989582 

.989553 

.939525 

.989497 

.989469 

.989441 

.989413 

.989385 

.989356 

.989328 

9.989300 

.989271 

.989243 

.989214 

.989186 

.989157 

.989128 

.989100 

.989071 

.989042 

9.989014 
.988985 
.98S956 
.988927 
.988898 
.988869 • 
.988840 
.988811 

. .988782 
.988753 
.988724 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

AT- 

AT 

AT 

AT 

AT 

AT 

AT 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.49 

.49 

9.327475 

.328095 

.328715 

.329334 

.329953 

.330570 

.331187 

.331803 

.332418 

.333033 

9.333646 

.334259 

.334871 

.335482 

.336093 

.336702 

.337311 

.337919 

.338527 

.339133 

9.339739 

.340344 

.340948 

.341552 

.342155 

.342757 

.343358 

.343958 

.344558 

.345157 

9.345755 

.346353 

.346949 

.347545 

.348141 

.348735 

.349329 

.349922 

.350514 

.351106 

9.351697 

.352287 

,352876 

.353465 

.354053 

.354640 

.355227 

.355813 

.356398 

.356982 

9.357566 

.358149 

.358731 

.359313 

.359893 

.360474 

.361053 

.361632 

.362210 

.362787 

.363364 

10.35 
10.33 
10.32 
10.31 
10.29 
10.28 
10.27 
• 10.25 
10.24 
10.23 

10.21 

10.20 

10.19 

10.17 

10.16 

10.15 

10.14 

10.12 

10.11 

10.10 

10.08 

10.07 

10.06 

10.05 

10.03 

10.02 

10.01 

10.00 

9.98 

9.97 

9.96 

9.95 

9.93 

9.92 

9.91 

9.90 

9.88 

9.87 

9.86 

9.85 

9.84 

9.82 

9.81 

9.80 

9.79 

9.78 

9.76 

9.75 

9.74 

9.73 

9.72 

9.70 

9.69 

9.68 

9.67 

9.66 

9.65 

9.63 

9.62 

9.61 

0.672525 
.671905 
.671285 
.670666 
.670047 
.669430 
.668813 
.663197 
.667582 
.666967 

0.666354 

.665741 

.665129 

.664518 

.663907 

.663298 

.662689 

.662081 

.661473 

.660867 

0.660261 

.659656 

.659052 

.658448 

.657845 

.657243 

.656642 

.656042 

.655442 

.654843 

0.654245 

.653647 

.653051 

.652455 

.651859 

.651265 

.650671 

.650078 

.649486 

.648894 

0.648303 

.647713 

.647124 

.646535 

.645947 

.645360 

.644773 

.644187 

.643602 

.643018 

0.642434 

.641851 

.641269 

.640687 

.640107 

.639526 

.638947 

.638368 

.637790 

.637213 

.636636 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

l 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


TT C 


1030 



































COSINES TANGENTS, AND COTANGENTS 


187 

IGfta 


13° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.352038 

.352635 

.353181 

.353726 

.354271 

.354815 

.355358 

.355901 

.356443 

.356984 

9.357524 

.353064 

.358603 

,359141 

.359678 

.360215 

.360752 

.361237 

.361822 

.362356 

9.362889 

.363422 

.363954 

.364485 

.365016 

.365546 

.366075 

.366604 

.367131 

.367659 

9.363185 
.36371l 
.369236 
.369761 
.370235 
.370303 
.371330 
.371852 
.372373 
.372894 

9.373414 

.373933 

.374452 

.374970 

.375487 

.376003 

.376519 

.377035 

.377549 

.378053 

9.378577 

.379089 

.379601 

.380113 

.330624 

.381134 

.381643 

.332152 

.332661 

.383168 

.383675 

9.11 

9.10 

9.09 

9.08 

9.07 

9.05 

9.04 

9.03 

9.02 

9.01 

8.99 

8.93 
8.97 
8.96 
8.95 

8.94 
8.92 
8.91 
8.90 
8.89 

8.88 

8.87 

8.86 

8.84 

8.83 

8.82 

8.81 

8.80 

8.79 

8.78 

8.76 

8.75 

8.74 

8.73 

8.72 

8.71 

8.70 

8.69 

8.63 
8.66 

8.65 

8.64 
8.63 
8.62 
8.61 
8.60 
8.59 
8.58 
8.57 
8.56 

8.55 

8.53 

8.52 

8.51 

8.50 

8.49 

8.48 

8.47 

8.46 

8.45 

9.988724 

.938695 

.933666 

.988636 

.988607 

.938578 

.938548 

.988519 

.988489 

.938460 

9.933430 

.933401 

.938371 

.933342 

.938312 

.938282 

.938252 

.938223 

.983193 

.983163 

9.938133 

.933103 

.938073 

.988043 

.938013 

.937933 

.937953 

.937922 

.937892 

.937862 

9.937832 

.937301 

.987771 

.937740 

.937710 

.987679 

.937649 

.987618 

.937588 

.937557 

9.987526 

.937496 

.987465 

.937434 

.987403 

.937372 

.937341 

.987310 

.987279 

.937248 

9.9S7217 

.937186 

.937155 

.937124 

.987092 

.937061 

.937030 

.986998 

.936967 

.986936 

.986904 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

9.363364 

.363940 

.364515 

.365090 

.365664 

.366237 

.366810 

.367332 

.367953 

.363524 

9.369094 

.369663 

•370232 

.370799 

.371367 

.371933 

.372499 

.373064 

.373629 

.374193 

9.374756 

.375319 

.375881 

.376442 

.377003 

.377563 

.378122 

.378631 

.379239 

.379797 

9.380354 

.330910 

.331466 

.332020 

.332575 

.383129 

.383682 

.384234 

.384786 

.335337 

9.385888 

.336438 

.386937 

.337536 

.383084 

.383631 

.389178 

.389724 

.390270 

.390315 

9.391360 

.391903 

.392447 

.392989 

.393531 

.394073 

.394614 

.395154 

.395694 

.396233 

.396771 

9.60 

9.59 

9.58 

9.57 

9.55 

9.54 

9.53 

9.52 

9.51 

9.50 

9.49 

9.48 

9.47 

9.45 

9.44 

9.43 

9.42 

9.41 

9.40 

9.39 

9.38 

9.37 

9.36 

9.35 

9.33 

9.32 

9.31 

9.30 

9.29 

9.23 

9.27 

9.26 

9.25 

9.24 
9.23 
9.22 
9.21 
9.20 
9.19 
9.18 

9.17 

9.16 

9.15 

9.14 

9.12 

9.11 

9.10 

9.09 

9.08 

9.07 

9.06 

9.05 

9.04 

9.03 

9.02 

9.01 

9.00 

8.99 

8.93 

8.97 

0.636636 

.636060 

.635485 

.634910 

.634336 

.633763 

.633190 

.632618 

.632047 

.631476 

0.630906 

.630337 

.62976S 

.629201 

.628633 

.628067 

.627501 

.626936 

.626371 

.625807 

0.625244 

.624631 

.624119 

.623558 

.622997 

.622437 

.621878 

.621319 

.620761 

.620203 

0.619646 

.619090 

.618534 

.617980 

.617425 

.616371 

.616318 

.615766 

.615214 

.614663 

0.614112 

.613562 

.613013 

.612464 

.611916 

.611369 

.610822 

.610276 

.609730 

.609185 

0.608640 

.608097 

.607553 

.607011 

.606469 

.605927 

.605386 

.604846 

.604306 

.603767 

.603229 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

23 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


7G3 


1030 





































im 

14° 


TABLE XIII. LOGARITHMIC SINES, 


165 a 


M. 

Sine. 

D. 1". 

Cosine. 

©. 1". 

Tang. 

D. 1". 

Co tang 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 
27 
23 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.383675 

.384182 

.384637 

.385192 

.385697 

.336201 

.336704 

.337207 

.337709 

.388210 

9.383711 
.339211 
.339711 
.390210 
.390708 
.391206 
.391703 
.392199 
.392695 
.393191 

9.393685 

.394179 

.394673 

.395166 

.395658 

.396150 

.396641 

.397132 

.397621 

.398111 

9.393600 

.399038 

.399575 

.400062 

.400549 

.401035 

.401520 

.402005 

.402489 

.402972 

9.403455 

.403933 

.404420 

.404901 

.405332 

.405862 

.406341 

.406820 

.407299 

.407777 

9.403254 

.403731 

.409207 

.409632 

.410157 

.410632 

.411106 

.411579 

.412052 

.412524 

.412996 

8.44 

8.43 

8.42 

8.41 

8.40 

8.39 

8.38 

8.37 

8.36 

8.35 

8.34 

8.33 

8.32 

8.31 

8.30 

8.29 

8.23 
8.27 
8.26 
8.25 

8.24 
8.23 
8.22 
8.21 
8.20 
8.19 
8.18 
8.17 
8.16 
8.15 

8.14 

8.13 

8.12 

8.11 

8.10 

8.09 

8.03 

8.07 

8.06 

8.05 

8.04 

8.03 

8.02 

8.01 

8.00 

7.99 

7.98 

7.97 

7.96 

7.96 

7.95 

7.94 

7.93 

7.92 

7.91 

7.90 

7.89 

7.88 

7.87 

7.86 

9.986904 

.986873 

.986341 

.986809 

.986778 

.936746 

.936714 

.986683 

.986651 

.986619 

9.986587 

.986555 

.936523 

.936491 

.986459 

.936427 

.986395 

.936363 

.986331 

.986299 

9.986266 

.936234 

.986202 

.986169 

.936137 

.936104 

.986072 

.986039 

.936007 

.985974 

9.935942 

.985909 

.985876 

.985843 

.935311 

.985778 

.985745 

.985712 

.985679 

.985646 

9.985613 

.985580 

.985547 

.985514 

.935480 

.985447 

.985414 

.985331 

.935347 

.935314 

9.985230 

.985247 

.985213 

.935180 

.935146 

.935113 

.935079 

.935045 

.935011 

.934978 

.934944 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

.56 

9.396771 

.397309 

.397846 

.398383 

.398919 

.399455 

.399990 

.400524 

.401058 

.401591 

9.402124 

.402656 

.403187 

.403718 

.404249 

.404778 

.405308 

.405836 

.406364 

.406892 

9.407419 

.407945 

.408471 

.408996 

.409521 

.410045 

.410569 

.411092 

.411615 

.412137 

9.412658 

.413179 

.413699 

.414219 

.414738 

.415257 

.415775 

.416293 

.416810 

.417326 

9.417842 

.418358 

.418873 

.419387 

.419901 

.420415 

.420927 

.421440 

.421952 

.422463 

9.422974 
.423484 
.423993 
.424503 
.425011 
.425519 
.426027 
.426534 
.427041 
.427547 
.428052 

8.96 

8.96 

8.95 

8.94 

8.93 

8.92 

8.91 

8.90 

8.89 

8.88 

8.87 

8.86 

8.85 

8.84 

8.83 

8.82 

8.81 

8.80 

8.79 

8.78 

8.77 

8.76 

8.75 

8.75 

8.74 

8.73 

8.72 

8.71 

8.70 

8.69 

8.68 

8.67 

8.66 

8.65 

8.65 

8.64 

8.63 

8.62 

8.61 

8.60 

8.59 

8.58 

8.57 

8.56 

8.56 

8.55 

8.54 

8.53 

8.52 

8.51 

8.50 

8.49 

8.49 

8.48 

8.47 

8.46 

8.45 

8.44 

8.43 

8.43 

0.603229 

.602691 

.602154 

.601617 

.601081 

.600545 

.600010 

.599476 

.598942 

.598409 

0.597876 

.597344 

.596813 

.596282 

.595751 

.595222 

.594692 

.594164 

.593636 

.593108 

0.592581 

.592055 

.591529 

.591004 

.590479 

.589955 

.589431 

.588908 

.588385 

.587863 

0.587342 

.586821 

.586301 

.585781 

.585262 

.584743 

.584225 

.583707 

.583190 

.582674 

0.582158 

.581642 

.581127 

.580613 

.580099 

.579585 

.579073 

.578560 

.578048 

.577537 

0.577026 

.576516 

.576007 

.575497 

.574989 

.574481 

.573973 

.573466 

.572959 

.572453 

.571948 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1" 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 




1040 




































COSINES, TANGENTS, AND COTANGENTS 


189 

15 C 164-0 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

ro 

60 

9.412996 

.413467 

.413938 

.414408 

.414878 

.415347 

.415815 

.416283 

.416751 

.417217 

9.417684 

.418150 

.418615 

.419079 

.419544 

.420007 

.420470 

.420933 

.421395 

.421857 

9.422318 

.422778 

.423238 

.423697 

.424156 

.424615 

.425073 

.425530 

.425987 

.426443 

9.426899 

.427354 

.427809 

.428263 

.428717 

.429170 

.429623 

.430075 

.430527 

.430978 

9.431429 

.431879 

.432329 

.432778 

.433226 

.433675 

.434122 

.434569 

.435016 

.435462 

9.435908 

.436353 

.436798 

.437242 

.437686 

.438129 

;438572 

.439014 

.439456 

.439897 

.440338 

7.85 

7.84 

7.84 

7.83 

7.82 

7.81 

7.80 

7.79 

7.78 

7.77 

7.76 

7.75 

7.75 

7.74 

7.73 

7.72 

7.71 

7.70 

7.69 

7.68 

7.67 

7.67 

7.66 

7.65 

7.64 

7.63 

7.62 

7.61 

7.61 

7.60 

7.59 

7.58 

7.57 

7.56 

7.55 

7.55 

7.53 

7.52 

7.52 

7.51 

7.50 

7.49 

7.49 

7.48 

7.47 

7.46 

7.45 

7.44 

7.44 

7.43 

7.42 

7.41 

7.40 

7.40 

7.39 

7.38 

7.37 

7.36 

7 36 
7.35 

9.984944 

.984910 

.984876 

.984842 

.984808 

.984774 

.984740 

.984706 

.984672 

.984638 

9.984603 

.984569 

.984535 

.984500 

.984466 

.984432 

.984397 

.984363 

.984328 

.984294 

9.934259 

.984224 

.984190 

.984155 

.984120 

.984085 

.984050 

.984015 

.983981 

.983946 

9.983911 

.983875 

.983840 

.983805 

.983770 

.983735 

.983700 

.983664 

.983629 

.983594 

9.983558 

.983523 

.983487 

.983452 

.983416 

.983381 

.983345 

.983309 

.983273 

.983238 

9.983202 

.983166 

.983130 

.983094 

.983058 

.983022 

.982986 

.982950 

.982914 

.982.878 

.982842 

.56 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

59 

.59 

.59 

.59 

.59 

.59 

.59 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

9.428052 

.428558 

.429062 

.429566 

.430070 

.430573 

.431075 

.431577 

.432079 

.432580 

9.433080 

.433580 

.434080 

.434579 

.435078 

.435576 

.436073 

.436570 

.437067 

.437563 

9.438059 

.438554 

.439048 

.439543 

.440036 

.440529 

.441022 

.441514 

.442006 

.442497 

9.442988 

.443479 

.443968 

.444458 

.444947 

.445435 

.445923 

.446411 

.446898 

.447384 

9.447870 

.448356 

.448841 

.449326 

.449810 

.450294 

.450777 

.451260 

.451743 

.452225 

9.452706 

.453187 

.453668 

.454148 

.454628 

.455107 

.455586 

.456064 

.456542 

.457019 

.457496 

8.42 

8.41 

8.40 

8.39 

8.38 

8.38 

8.37 

8.36 

8.35 

8.34 

8.33 
8.33 
8.32 
8.31 
8.30 
8.29 
■ 8.28 
8.28 
8.27 
8.26 

8.25 

8.24 

8.24 

8.23 

8.22 

8.21 

8.20 

8.20 

8.19 

8.18 

8.17 

8.16 

8.16 

8.15 

8.14 

8.13 

8.13 

8.12 

8.11 

8.10 

8.09 

8.09 

8.08 

8.07 

8.06 

8.06 

8.05 

8.04 

8.03 

8.03 

8.02 

8.01 

8.00 

8.00 

7.99 

7.98 

7.97 

7.97 

7.96 

7.95 

0.571948 

.571442 

.570938 

.570434 

.569930 

.569427 

.568925 

.568423 

.567921 

.567420 

0.566920 

.566420 

.565920 

.565421 

.564922 

.564424 

.563927 

.5/33430 

.562933 

.562437 

0.561941 

.561446 

.560952 

.560457 

.559964 

.559471 

.558978 

.558486 

.557994 

.557503 

0.557012 

.556521 

.556032 

.555542 

.555053 

.554565 

.554077 

.553589 

.553102 

.552616 

0.552130 

.551644 

.551159 

.550674 

.550190 

.549706 

.549223 

.548740 

.548257 

.547775 

0.547294 

.546813 

.546332 

.545852 

.545872 

.544893 

.544414 

.543936 

.543458 

.542981 

.542504 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

8 
2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


1050 74 = 


































TABLE Xlli. LOGARITHMIC SINES, 


163 c 


190 


160 


M. 

Sine. 

D 1". 

Cosine. 1 

D. 1 ,( . 

Tang. 

D. 1". 

Cotang. 

M. 

0 

9.440333 

7.34 

7.33 

7.32 

7.31 

7.31 

7.30 

7.29 

9.932842 

.60 

.60 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

,62 

.62 

.62 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

9.457496 

7.94 

7 94 

0.542504 

60 

1 

.440778 

.982805 

.457973 

.542027 

59 

2 

.441218 

.932769 

.458449 

7.93 

7.92 

7.91 

7.91 

7.90 

7.89 

7.83 

7.83 

7.87 

7.86 

7.86 

7.85 

7.84 
7.83 
7.83 
7.82 
7.81 
7.81 

7.30 

7.79 

7.78 

7.78 

7.77 

7.76 

7.76 

rr 

.541551 

58 

3 

.441653 

.982733 

.458925 

.541075 

57 

4 

.442096 

.982696 

.459400 

.540600 

56 

5 

.442535 

.932660 

.459375 

.540125 

55 

6 

.442973 

.932621 

.460349 

.539651 

54 

7 

.443410 

.982587 

.460323 

.539177 

53 

8 

.443847 

7.23 
7.27 
7.27 

7.26 

7.25 

7.24 
7.24 
7.23 

.982551 

.461297 

.538703 

52 

9 

10 

.444234 

9.444720 

.982514 

9.982477 

.461770 

9.462242 

.533230 

0.537758 

51 

50 

] 1 

.445155 

.982441 

.462715 

.537285 

49 

12 

.445590 

.982404 

.463186 

.536814 

48 

13 

.446025 

.982367 

.463658 

.536342 

47 

14 

.446459 

.982331 

.464128 

.535872 

46 

15 

.446393 

.982294 

.464599 

.535401 

45 

16 

.447326 

7.22 

7.21 

.982257 

.465069 

.534931 

44 

17 

.447759 

.932220 

.465539 

.534461 

43 

18 

.443191 

7.20 

.932183 

.466008 

.533992 

42 

19 

20 

.443623 

9.449054 

7.20 

7.19 

7.18 

7.17 

7.17 

7.16 

7.15 

.932146 

9.932109 

.466177 

9.466945 

.533523 

0.533055 

41 

40 

21 

' .449435 

.932072 

.467413 

.532587 

39 

22 

.419915 

.932035 

.467880 

.532120 

38 

23 

.450345 

.931993 

.463347 

.531653 

37 

24 

.450775 

.931961 

.468814 

.5311S6 

36 

25 

.451204 

.931924 

.469280 

.530720 

35 

26 

.451632 

7.14 

.931886 

.469746 

.530254 

34 

27 

.452060 

7.13 

.981849 

.470211 

.529789 

33 

28 

.452488 

7.13 

.981812 

.470676 

7.74 

7.74 

7.73 

7.72 

7.71 

7.71 

7.70 

7.69 

7.69 

7.63 
7.67 
7.67 

7.66 

7.65 

7.65 

7.64 
7.63 
7.63 
7.62 
7.61 
7.61 
7.60 

7.59 

7.59 

7.58 

7.57 

7.57 

7.56 

7.55 

7 55 
7.54 
7.53 

.529324 

32 

29 

30 

.452915 

9.453342 

7.12 
7.11 

7.10 

7.10 

.931774 

9.931737 

.471141 

9.471605 

.523859 

0.523395 

31 

30 

31 

.453763 

.981700 

.472069 

.527931 

29 

32 

.454194 

.981662 

.472532 

.527463 

28 

33 

.454619 

7.09 

.981625 

.472995 

.527005 

27 

34 

.455044 

7.03 

.981587 

.473457 

.526543 

26 

35 

.455469 

7.07 

7.07 

7.06 

.931549 

.473919 

.526031 

25 

36 

.455393 

.981512 

.474331 

.520619 

24 

37 

.456316 

.981474 

.474842 

.525158 

23 

38 

.456739 

7.05 

7.04 

7.04 

7.03 

7.02 

.931436 

.475303 

.524697 

22 

39 

40 

.457162 

9.457584 

.931399 

9.981361 

.475763 

9.476223 

.524237 

0.523777 

21 

20 

41 

.458006 

.981323 

.476633 

.523317 

19 

42 

.453427 

.981235 

.477142 

.522858 

18 

43 

.458348 

7.01 

.931247 

.477601 

.522399 

17 

44 

.459268 

7.01 

.981209 

.478059 

.521941 

16 

45 

.459688 

7.00 

.981171 

.478517 

.521483 

15 

46 

.460103 

6.99 

.981133 

.478975 

.521025 

14 

47 

.460527 

6.98 

6.98 

6.97 

6.96 

6.96 

6.95 

6.94 

6.93 

6.93 

6.92 

6.91 

6.90 

6.90 

6.89 

.981095 

.479432 

.520563 

13 

48 

.460946 

.981057 

.479389 

.520111 

12 

49 

50 

.461364 

9.461782 

.981019 

9.930981 

.430345 

9.480801 

.519655 

0.519199 

11 

10 

51 

.462199 

.930942 

.481257 

.518743 

9 

52 

.462616 

.930904 

.481712 

.518288 

8 

53 

.463032 

.980366 

.482167 

.517833 

7 

54 

.463448 

.980327 

.482621 

.517379 

6 

55 

.463364 

.930789' 

.483075 

.516925 

5 

56 

.464279 

.980750 

.483529 

.516471 

4 

I 5f 

.464694 

.930712 

.483982 

.516018 

3 

' 58 

.465108 

.980673 

.484435 

.515565 

2 

59 

.465522 

.930635 

.434837 

.515113 

1 

| 60 

.465935 

.980596 

.485339 

.514661 

0 

!"• 

| Cosine. 

1 D. 1”. 

Sine. 

D. 1". 

1 Cotang. 

D. 1". 

Tang. 

M. 



































191 

1G33 


COSINES, TANGENTS, AND COTANGENTS. 

I To 


M 

Sine. 

D 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

It 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 
23 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.465935 

.466343 

.466761 

.467173 

.467585 

.467996 

.463407 

.463317 

.469227 

.469637 

9.470046 

.470455 

.470363 

.471271 

.471679 

.472036 

.472492 

.472393 

.473304 

.473710 

9.474115 

.474519 

.474923 

.475327 

.475730 

.476133 

.476536 

.476933 

.477340 

.477741 

9.478142 

.473542 

.478942 

.479342 

.479741 

.480140 

.480539 

.480937 

.481334 

.481731 

9.432128 

.482525 

.482921 

.483316 

.483712 

.434107 

.434501 

.434395 

.435239 

.485632 

9.486075 

.486467 

.486860 

.437251 

.487643 

.488034 

.488424 

.488814 

.489204 

.489593 

.439932 

6 88 
6.88 
6.87 
6.86 
6.85 
6.85 
6.84 
6.83 
6.83 
6.82 

6.81 

6.81 

6.80 

6.79 

6.78 

6.78 

6.77 

6.76 

6.76 

6.75 

6.74 

6.74 

6.73 

6.72 

6.72 

6.71 

6.70 

6.69 

6.69 

6.63 

6.67 

6.67 

6.66 

6.65 

6.65 

6.64 
6.63 
6.63 
6.62 
6.61 

6.61 

6.60 

6.59 

6.59 

6.58 

6.57 

6.57 

6.56 

6.55 

6.55 

6.54 

6.54 

6.53 

6.52 

6.52 

6.51 

6.50 

6.50 

6.49 

6.48 

9.930596 

.930558 

.930519 

.930480 

.930442 

.930403 

.930364 

.930325 

.930286 

.930247 

9.980208 

.980169 

.930130 

.980091 

.930052 

.930012 

.979973 

.979934 

.979395 

.979855 

9.979316 

.979776 

.979737 

.979697 

.979653 

.979618 

.979579 

.979539 

.979499 

.979459 

9.979420 

.979330 

.979340 

.979300 

.979260 

.979220 

.979180 

.979140 

.979100 

.979059 

9.979019 

.978979 

.978939 

.978398 

.978858 

.978817 

.978777 

.973737 

.978696 

.978655 

9.978615 

.978574 

.978533 

.978493 

.978452 

.978411 

.978370 

.978329 

.978238 

.978247 

.978206 

.64 

.64 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.68 

.63 

.63 

.68 

.63 

.63 

.68 

.68 

.63 

.68 

.68 

.63 

.68 

9.485339 

.485791 

.486242 

.486693 

.487143 

.437593 

.488043 

.483492 

.488941 

.489390 

9.489833 

.490286 

.490733 

.491180 

.491627 

.492073 

.492519 

.492965 

.493410 

.493354 

9.494299 

.494743 

.41)5186 

.495630 

.496073 

.496515 

.496957 

.497399 

.497.841 

.493232 

9.498722 

.499163 

.499603 

.500042 

.500431 

.500920 

.501359 

.501797 

.502235 

.502672 

9.503109 

.503546 

.503932 

.504418 

.504854 

.505239 

.505724 

.506159 

.506593 

.507027 

9.507460 

.507893 

.503326 

.503759 

.509191 

.509622 

.510054 

.510485 

.510916 

.511346 

.511776 

7.53 

7.52 

7.51 

7.51 

7.50 

7.50 

7.49 

7.48 

7.48 

7.47 

7.46 

7.46 

7.45 

7.44 

7.44 

7.43 

7.43 

7.42 

7.41 

7.41 

7.40 

7.39 

7.39 

7.38 

7.33 
7.37 
7.36 
7.36 
7.35 

7.34 

7.34 

7.33 

7.33 

7.32 

7.31 

7.31 

7.30 

7.30 

7.29 

7.28 

7.28 

7.27 

7.27 

7.26 

7.25 

7.25 

7.24 

7.24 

7.23 

7.23 

7.22 

7.21 

7.21 

7.20 

7.20 

7.19 

7.18 

7.13 

7.17 

7.17 

0.514661 

.514209 

.513758 

.513307 

.512857 

.512407 

.511957 

.511503 

.511059 

.510610 

0.510162 

.509714 

.509267 

.508820 

.503373 

.507927 

.507481 

.507035 

.506590 

.506146 

0.505701 

.505257 

.504814 

.504370 

.503927 

.503435 

.503043 

.502601 

.502159 

.501718 

0.501278 

.500337 

.500397 

.499958 

.499519 

.499080 

.498641 

.493203 

.497765 

.497328 

0.496891 

.496454 

.496018 

.495582 

.495146 

.494711 

.494276 

.493841 

.493407 

.492973 

C1492540 

.492107 

.491674 

.491241 

.490809 

.490378 

.489946 

.489515 

.489084 

.488654 

.488224 

60 

59 

53 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

23 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 1 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


75SC 


ior^ 

































192 

18 ^ 


TABLE XIII. LOGARITHMIC SINES 


1G2< 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 
27 
23 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 
4S 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.489932 

.490371 

.490759 

.491147 

.491535 

.491922 

.492303 

.492695 

.493081 

.493466 

9.493851 

.494236 

.494621 

.495005 

.495383 

.495772 

.496154 

.496537 

.496919 

.497301 

9.497632 

.493064 

.498444 

.493825 

.499204 

.499584 

.499963 

.500342 

.500721 

.501099 

9.501476 

.501854 

.502231 

.502607 

.502984 

.503360 

.503735 

.504110 

.504485 

.504860 

9.505234 

.505603 

.505981 

.506354 

.506727 

.507099 

.507471 

.507843 

.503214 

.508585 

9.508956 

.509326 

.509696 

.510065 

.510434 

.510803 

.511172 

.511540 

.511907 

.512275 

.512642 

6.48 

0.47 

6.46 

6.46 

6.45 

6.45 

6.44 

6.43 

6.43 

6.42 

6.41 

6.41 

6.40 

6.39 

6.39 

6.33 

6.33 
6.37 
6.36 
6.36 

6.35 

6.34 
6.34 
6.33 
6.33 
6.32 
6.31 
6.31 
6.30 
6.30 

6.29 

6.23 

6.23 
6.27 
6.27 
6.26 
6.25 
6.25 

6.24 
6.24 

6.23 

6.22 

6.22 

6.21 

6.21 

6.20 

6.19 

6.19 

6.18 

6.18 

6.17 

6.16 

6.16 

6.15 

6.15 

6.14 

6.14 

6.13 

6.12 

6.12 

9.978206 

.978165 

.978124 

.978033 

.978042 

.978001 

.977959 

.977918 

.977877 

.977835 

9.977794 

.977752 

.977711 

.977669 

.977628 

.977536 

.977544 

.977503 

.977461 

.977419 

9.977377 

.977^35 

.977293 

.977251 

.977209 

.977167 

.977125 

.977083 

.977041 

.976999 

9.976957 

.976914 

.976372 

.976830 

.976787 

.976745 

.976702 

.976660 

.976617 

.976574 

9.976532 

.976489 

.976446 

.976404 

.976361 

.976318 

.976275 

.976232 

.976189 

.976146 

9.976103 

.976060 

.976017 

.975974 

.975930 

.975387 

.975844 

.975800 

.975757 

.975714 

.975670 

.63 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

9.511776 

.512206 

.512635 

.513064 

.513493 

.513921 

.514349 

.514777 

.515204 

.515631 

9.516057 

.516484 

.516910 

.517335 

.517761 

.518186 

.518610 

.519034 

.519458 

.619882 

9.520305 

.520728 

.521151 

.521573 

.521995 

.522417 

.522838 

.523259 

.523680 

.524100 

9.524520 

.524940 

.525359 

.525778 

.526197 

.526615 

.527033 

.527451 

.527863 

.528235 

9.528702 

.529119 

.529535 

.529951 

.530366 

.530781 

.531196 

.531611 

.532025 

.532439 

9.532353 

.533266 

.533679 

.534092 

.531504 

.534916 

.535328 

.535739 

.536150 

.536561 

.536972 

7.16 

7.16 

7.15 

7.14 

7.14 

7.13 

7.13 

7.12 

7.12 

7.11 

7.10 

7.10 

7.09 

7.09 

7.03 

7.08 

7.07 

7.07 

7.06 

7.05 

7.05 

7.04 

7.04 

7.03 

7.03 

7.02 

7.02 

7.01 

7.01 

7.00 

6.99 

6.99 

6.95 
6.98 
6.97 
6.97 

6.96 
6.96 
6.95 
6.95 

6.94 

6.94 

6.93 

6.93 

6.92 

6.91 

6.91 

6.90 

6.90 

6.89 

6.89 

6.88 

6.88 

6.87 

6.87 

6.86 

6.86 

6.85 

6.85 

6.S4 

0.488224 

.487794 

.487365 

.486936 

.486507 

.486079 

.485651 

.485223 

.484796 

.484369 

0.483943 

.483516 

.483090 

.482665 

.482239 

.481814 

.481390 

.480966 

.480542 

.480118 

0.479695 

.479272 

.478849 

.478427 

.478005 

.477533 

.477162 

.476741 

.476320 

.475900 

0.475480 

.475060 

.474641 

.474222 

.473803 

.473335 

.472967 

.472549 

.472132 

.471715 

0.471298 

.470881 

.470465 

.470049 

.469634 

.469219 

.468804 

.468389 

.467975 

.467561 

0.467147 

.466734 

.466321 

.465903 

.465496 

.465084 

.464672 

.464261 

.463350 

.463439 

.463028 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 
• 39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


108 3 7ia 


































COSINES, TANGENTS, AND COTANGENTS 


193 


IIP 160 ° 


M. 

Sine. 

D. 1”. 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Co tang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.512642 

.513009 

.513375 

.513741 

.514107 

.514472 

.514837 

.515202 

.515566 

.515930 

9.516294 

.516657 

.517020 

.517332 

.517745 

.518107 

.518468 

.518829 

.519190 

.519551 

9.519911 

.520271 

.520631 

.520990 

.521349 

.521707 

.522066 

.522424 

.522781 

.523138 

9.523495 

.523352 

.524208 

.524564 

.524920 

.525275 

.525630 

.525934 

.526339 

.526693 

9.527046 

.527400 

.527753 

.523105 

.528458 

.528810 

.529161 

.529513 

.529364 

.530215 

9.530565 

.530915 

.531265 

.531614 

.531963 

.532312 

.532661 

.533009 

.533357 

.533704 

.531052 

6.11 

6.11 

6.10 

6.09 

6.09 

6.03 

6.03 

6.07 

6.07 

6.06 

6.05 

6.05 

6.04 

6.04 

6.03 

6.03 

6.02 

6.02 

6.01 

6.00 

6.00 

5.99 

5.99 

5.98 

5.98 

5.97 

5.97 

5.96 

5.95 

5.95 

5.94 

5.94 

5.93 

5.93 

5.92 

5.92 

5.91 

5.90 

5.90 

5.89 

5.89 

5.88 

5.88 

5.87 

5.87 

5.86 

5.86 

5.85 

5.85 

5.84 

5.83 

5.83 

5.82 

5.82 

5.81 

5.81 

5.80 

5.80 

5.79 

5.79 

9.975670 

.975627 

.975583 

.975539 

.975496 

.975452 

.975408 

.975365 

.975321 

.975277 

9.975233 

.975189 

.975145 

.975101 

.975057 

.975013 

.974969 

.974925 

.974880 

.974336 

9.974792 

.974748 

.974703 

.974659 

.974614 

.974570 

.974525 

.974481 

.974436 

.974391 

9.974347 

.974302 

.974257 

.974212 

.974167 

.974122 

.974077 

.974032 

.973987 

.973942 

9.973397 
.973852 
.973307 
.973761 
.973716 
.973671 
.973625 
.973530 
.973535 
.973489 

9.973444 

.973393 

.973352 

.973307 

.973261 

.973215 

.973169 

.973124 

.973078 

.973032 

.972986 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.77 

.77 

9.536972 

.537382 

.537792 

.538202 

.538611 

.539020 

.539429 

.539837 

.540245 

.540653 

9.541061 

.541468 

.541875 

.542231 

.542688 

.543094 

.543499 

.543905 

.544310 

.544715 

9.545119 

.545524 

.545923 

.546331 

.546735 

.547138 

.547540 

.547943 

.548345 

.548747 

9.549149 

.549550 

.549951 

.550352 

.550752 

.551153 

.551552 

.551952 

.552351 

.552750 

9.553149 

.553548 

.553946 

.554344 

.554741 

.555139 

.555536 

.555933 

.556329 

.556725 

9.557121 

.557517 

.557913 

.558308 

.558703 

.559097 

.559491 

.559885 

.560279 

.560673 

.561066 

6.84 

6.83 

6. S3 
6.82 
6.82 
6.81 
6.81 
6.80 
6.80 
6.79 

6.79 

6.78 

6.78 

6.77 

6.77 

6.76 

6.76 

6.75 

6.75 

6.74 

6.74 

6.73 

6.73 

6.72 

6.72 

6.71 

6.71 

6.70 

6.70 

6.69 

6.69 

6.63 
6.68 
6.67 
6.67 
6.67 
6.66 
6.66 
6.65 
6.65 

6.64 
6.64 
6.63 
6.63 
6.62 
6.62 
6.61. 
6.61 
6.60 
6.60 

6.59 

6.59 

6.59 

6.58 

6.58 

6.57 

6.57 

6.56 

6.56 

6.55 

0.463028 

.462618 

.462208 

.461798 

.461389 

.460980 

.460571 

.460163 

.459755 

.459347 

0.458939 

.458532 

.458125 

.457719 

.457312 

.456906 

.456501 

.456095 

.455690 

.455285 

0.454881 
■ .454476 
.454072 
.453669 
.453265 
.452362 
.452460 
.452057 
.451655 
.451253 

0.450851 

.450450 

.450049 

.449648 

.449248 

.448847 

.448448 

.448048 

.447649 

.447250 

0.446851 

.446452 

.446054 

.445656 

.445259 

.444861 

.444464 

.444067 

.443671 

.443275 

0.442879 

.442483 

.442087 

.441692 

.441297 

.440903 

.440509 

.440115 

.439721 

.439327 

.438934 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

23 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". | 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. | 

M. 


70 = 


1090 




































194 

eoo 


TABLE XIII. LOGARITHMIC SINES, 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

0 

9.534052 

§.78 

5.78 

5.77 

5.77 

5.76 

5.76 

5.75 

5.75 

5.74 

5.74 

9.972986 


9.561066 

1 

.534399 

.972940 

. / / 
.77 
.77 
.77 
.77 

.561459 

2 

.534745 

.872894 

.561851 

3 

.535092 

.972348 

.562244 

4 

.535438 

.972802 

.562636 

5 

.535783 

.972755 

.563028 

6 

.536129 

.972709 

. / / 
.77 
.77 
.77 
.77 

.56:3419 

7 

.536474 

.972663 

.563811 

8 

.536818 

.972617 

.564202 

9 

.537163 

.972570 

.564593 

10 

9.537507 

5.73 

5.73 

5.72 

5.71 

5.71 

5.70 

5.70 

5.69 

5.69 

5.68 

9.972524 


9.5649S3 

11 

.537851 

.972478 

. / / 
.-77 
.78 
.78 
.78 
.78 
.78 
.78 
.78 
.78 

.565373 

12 

.538194 

.972431 

.565763 

13 

.538538 

.972385 

.566153 

14 

.538880 

.972338 

.566542 

15 

.539223 

.972291 

.566932 

16 

.539565 

.972245 

.567320 

17 

.539907 

.972198 

.567709 

18 

.540249 

.972151 

.568098 

19 

.540590 

.972105 

.568486 

20 

9.540931 

5.68 

5.67 

5.67 

5.66 

5.66 

5.65 

5.65 

5.64 

5.64 

5.63 

9.972058 

.78 

7 ( Q 

9.568873 

21 

.541272 

.972011 

.569261 

22 

.541613 

.971964 

.78 

.78 

.78 

.78 

.78 

.79 

.79 

.79 

.569648 

23 

.541953 

.971917 

.570035 

24 

.542293 

.971870 

.570422 

25 

.542632 

.971823 

.570809 

26 

.542971 

.971776 

.571195 

27 

.543310 

.971729 

.571581 

28 

.543649 

.971682 

.571967 

29 

.543987 

.971635 

.572352 

30 

9.544325 

5.63 

5.62 

5.62 

5.61 

5.61 

5.60 

5.60 

5.59 

5.59 

5.58 

9.971588 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

9.572738 

31 

.544663 

.971540 

.573123 

32 

.545000 

.971493 

.573507 

33 

.545338 

.971446 

.573892 

34 

.545674 

.971398 

.574276 

35 

36 

.546011 
' .546347 

.971351 

.971303 

.574660 

.575044 

37 

.546683 

.971256 

.575427 

38 

.547019 

.971208 

.575810 

39 

.547354 

.971161 

.576193 

40 

9.547689 

5.58 

5.57 

5.57 

5.56 

5.56 

5.55 

5.55 

5.55 

5.54 

5.54 

9.971113 

.79 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

9.576576 

41 

.548024 

.971066 

.576959 

42 

.548359 

.971018 

.577341 

43 

.548693 

.970970 

.577723 

44 

.549027 

.970922 

.578104 

45 

46 

47 

.549360 

.549693 

.550026 

.970874 

.970827 

.970779 

.578486 

.578867 

.579248 

48 

.550359 

.970731 

.579629 

49 

.550692 

.970683 

.580009 

50 

9.551024 

5.53 

5.53 

5.52 

5.52 

5.51 

5.51 

5.50 

5.50 

5.49 

5.49 

9.970635 

.80 

.80 

.80 

.80 

.80 

.81 

.81 

.81 

.81 

.81 

9.580389 

51 

.551356 

.970586 

.580769 

52 

.551687 

.970538 

.581149 

53 

.552018 

.970490 

.581528 

54 

.552349 

.970442 

.5S1907 

55 

.552680 

.970394 

.582286 

56 

.553010 

.970345 

.582665 

57 

.553341 

.970297 

.583044 

58 

.553670 

.970249 

.583422 

59 

.551000 

.970200 

.583800 

60 

.554329 

.970152 

.584177 

M. 

Cosine. 

D. V-. 

Sine. 

D. 1". 

Cotang. 


1593 

I 


D. 1«. 


6.55 

6.54 

6.54 

6.54 

6.53 

6.53 

6.52 

6.52 

6.51 

6.51 

6.50 

6.50 

6.50 

6.49 

6.49 

6.48 

6.48 

6.47 

6.47 

6.46 

6.46 

6.46 

6.45 

6.45 

6.44 

6.44 

6.43 

6.43 

6.43 

6.42 

6.42 

6.41 

6.41 

6.40 

6.40 

6.40 

6.39 

6.39 

6.38 

6.38 

6.37 

6.37 

6.37 

6.36 

6.36 

6.35 

6.35 

6.34 

6.34 

6.34 

6.33 

6.33 

6.32 

6.32 

6.32 

6.31 

6.31 

6.30 

6.30 

6.30 


D. 1". 


Cotang. 


0.438934 

.438541 

.438149 

.437756 

.437364 

.436972 

.436581 

.436189 

.435798 

.435407 

0.435017 

.434627 

.434237 

.433847 

.433458 

.433068 

.432680 

.432291 

.431902 

.431514 

0.431127 

.430739 

.430352 

.429965 

.429578 

.429191 

.428805 

.428419 

.428033 

.427648 

0.427262 

.426877 

.426493 

.426108 

.425724 

.425340 

.424956 

.424573 

.424190 

.423807 

0.423424 

.423041 

.422659 

.422277 

.421896 

.421514 

.421133 

.420752 

.420371 

.419991 

0.419611 

.419231 

.418851 

.418472 

.418093 

.417714 

.417335 

.416956 

.416578 

.416200 

.415823 


M. 


Tang. 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


M. 


I1QO 








































210 


COSINES, TANGENTS, AND COTANGENTS 


195 

1580 


M. 

Sine. 

D. I' 1 . 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

o 

O 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.554329 

.554658 

.554987 

.555315 

.555643 

.555971 

.556299 

.556626 

.556953 

.557280 

9.557603 

.557932 

.558258 

.558583 

.558909 

.559234 

.559558 

.559883 

.560207 

.560531 

9.560855 

.561178 

.561501 

.561824 

.562146 

.562463 

.562790 

.563112 

.563433 

.563755 

9.564075 

.564396 

.564716 

.565036 

.565356 

.565676 

.565995 

.566314 

.566632 

.566951 

9.567269 

.567587 

.567904 

.568222 

.563539 

.568856 

.569172 

.569488 

.569804 

.570120 

9.570435 

.570751 

.571066 

.571380 

.571695 

.572009 

.572323 

.572636 

572950 

.573263 

.573575 

5.48 

5.48 

5.47 

5.47 

5.46 

5.46 

5.45 

5.45 

5.44 

5.44 

5.44 

5.43 

5.43 

5.42 

5.42 

5.41 

5.41 

5.40 

5.40 

5.39 

5.39 

5.38 

5.38 

5.37 

5.37 

5.37 

5.36 

5.36 

5.35 

5.35 

5.34 

5.34 

5.33 

5.33 

5.32 

5.32 

5.32 

5.31 

5.31 

5.30 

5.30 

5.29 

5.29 

5.23 

5.28 

5.23 
5.27 
5.27 
5.26 
5.26 

5.25 

5.25 

5.24 
5.24 
5.24 
5.23 
5.23 
5.22 
5.22 
5.21 

9.970152 

.970103 

.970055 

.970006 

.969957 

.969909 

.969860 

.969811 

.969762 

.969714 

9.969665 

.969616 

.969567 

.969518 

.969469 

.969420 

.969370 

.969321 

.969272 

.969223 

9.969173 

.969124 

.969075 

.969025 

.968976 

.968926 

.968377 

.963827 

.968777 

.968728 

9.968678 

.968628 

.968578 

.968528 

.96S479 

.968429 

.968379 

.963329 

.968278 

.968228 

9.963178 
.968128 
.968078 
.968027 
' .967977 
.967927 
.967876 
.967826 
.967775 
.967725 

9.967674 

.967624 

.967573 

.967522 

.967471 

.967421 

.967370 

.967319 

.967268 

.967217 

.967166 

.81 

.81 

.81 

.81 

81 

.81 

.81 

.81 

.81 

.81 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.82 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.83 

.84 

.84 

84 

.84 

.84 

.84 

.84 

.84 

.84 

.84 

.84 

.84 

.84 

.84 

.85 

.85 

.85 

.85 

.85 

.85 

.85 

.85 

9.584177 

.584555 

.584932 

.585309 

.585686 

.586062 

.586439 

.586815 

.587190 

.587566 

9.587941 

.588316 

.588691 

.589066 

.589440 

.589814 

.590188 

.590562 

.590935 

.591308 

9.591631 

.592054 

.592426 

.592799 

.593171 

.593542 

.593914 

.594285 

.594656 

.595027 

9.595398 

.595763 

.596138 

.596508 

.596878 

.597247 

.597616 

.597985 

.598354 

.598722 

9.599091 

.599459 

.599827 

.600194 

.600562 

.600929 

.601296 

.601663. 

.602029 

.602395 

9.602761 

.603127 

.603493 

.603858 

.604223 

.604588 

.604953 

.605317 

.605682 

.606046 

.606410 

6.29 

6.29 

6.28 

6.28 

6.28 

6.27 

6.27 

6.26 

6.26 

6.26 

6.25 

6.25 

6.24 

6.24 

6.24 

6.23 

6.23 

6.22 

6.22 

6.22 

6.21 

6.21 

6.20 

6.20 

6.20 

6.19 

6.19 

6.18 

6.18 

6.18 

6.17 

6.17 

6.16 

6.16 

6.16 

6.15 

6.15 

6.15 

6.14 

6.14 

6.13 

6.13 

6.13 

6.12 

6.12 

6.12 

6.11 

6.11 

6.10 

6.10 

6.10 

6.09 

6.09 

6.09 

6.08 

6.08 

6.07 

6.07 

6.07 

6.06 

0.415823 
.415445 
.415068 
.414691 
.414314 
.4 i 3938 
.413561 
.413185 
.412810 
.412434 

0.412059 

.411684 

.411309 

.410934 

.410560 

.410186 

.409812 

.409438 

.409065 

.408692 

0.408319 

.407946 

.407574 

.407201 

.406829 

.406458 

.406086 

.405715 

.405344 

.404973 

0.404802 

.404232 

.403862 

.403492 

.403122 

.402753 

.402384 

.402015 

.401646 

.401278 

0.400909 
.400541 
.400173 
.399806 
.399438 
.399071 
.398704 
.398337 
.397971 . 
.397605 

0.397239 
.396873' 
.396507 
.396142 
.395777 
.395412 
.395047 
.394683 
.394318 
.393954 
.393590 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


zss. 


111P 


G8« 











































TABLE XIII. LOGARITHMIC SINES, 


1573 


196 

933 


M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

6.06 

6.06 

6.05 

6.05 

6.05 

6.04 

6.04 

6.03 

6.03 

6.03 

6.02 

6.02 

6.02 

6.01 

6.01 

6.01 

6.00 

6.00 

6.00 

5.99 

5.99 

5.98 

5.98 

5.93 
5.97 
5.97 
5.97 
5.96 
5.96 
5.96 

5.95 

5.95 

5.95 

5.94 
5.94 
5.94 
5.93 
5.93 
5.93 
5.92 

5.92 

5.92 

5.91 

5.91 

5.91 

5.90 

5.90 

5.90 

5.89 

5.89 

5.89 

5.88 

5.88 

5.88 

5.87 

5.87 

5.87 

5.86 

5.86 

5.86 

Cotang. 

M. 

9.573575 

.573883 

.574200 

.574512 

.574824 

.575136 

.575447 

.575758 

.576069 

.576379 

9.576639 

.576999 

.577309 

.577618 

.577927 

.578236 

.578545 

.578853 

.579162 

.579170 

9.579777 
.580035 
.580392 
.580699 
.5S1005 
.581312 
.531618 
.581924 
.582229 
.582535 

9.582340 

.533145 

.533449 

.533754 

.581058 

.534361 

.534665 

.534963 

.585272 

.585574 

9.535377 

.586179 

.536482 

.536783 

.587085 

.537336 

.587638 

.537939 

.588239 

.533590 

9.538890 

.539190 

.589439 

.539789 

.590038 

.590337 

.590636 

.590934 

.591232 

.591530 

.591878 

5.21 

5.20 

5.20 

5.20 

5.19 

5.19 

5.18 

5.18 

5.17 

5.17 

5.17 

5.16 

5.16 

5.15 

5.15 

5.14 

5.14 

5.14 

5.13 

5.13 

5.12 

5.12 

5.11 

5.11 

5.11 

5.10 

5.L0 

5.09 

5.09 

5.09 

5.03 

5.08 

5.07 

5.07 

5.06 

5.06 

5.06 

5.05 

5.05 

5.04 

5.04 

5.04 

5.03 

5.03 

5.02 

5.02 

5.01 

5.01 

5.01 

5.00 

5.00 

4.99 

4.99 

4.99 

4.93 

4.93 

4.97 

4.97 

4.97 

4.96 

9.967166 
.967115 
.967061 
.967013 
.966961 
.966910 
.966359 
.966303 
.966756 
.966705 

9.966653 

.966602 

.966550 

.966499 

.966447 

.966395 

.966344 

.966292 

.966240 

.968183 

9.966135 

.966035 

.966033 

.965981 

.965929 

.965376 

.965324 

.965772 

.965720 

.965663 

9.965615 

.965563 

.965511 

.965458 

.965406 

.965353 

.965301 

.965248 

.965195 

.965143 

9.965090 

.965037 

.964934 

.964931 

.964379 

.964826 

.964773 

-.964720 

.964666 

.961613 

9.964560 

.964507 

.964454 

.964400 

.964347 

.964294 

.964240 

.964187 

.964133 

.964080 

.961026 

.85 

.85 

.85 

.85 

.85 

.85 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.83 

.88 

.83 

.83 

.88 

.83 

.83 

.88 

.83 

.88 

.83 

.83 

.88 

.88 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

9.606110 

.606773 

.607137 

.607500 

.607863 

.603225 

.608588 

.608950 

.609312 

.609674 

9.610036 

.610397 

.610759 

.611120 

.611480 

.611841 

.612201 

.612561 

.612921 

.613231 

9.613641 

.614000 

.614359 

.614718 

.615077 

.615435 

.615793 

.616151 

.616509 

.616367 

9.617224 

.617582 

.617939 

.618295 

.618652 

.619008 

.619364 

.619720 

.620076 

.620432 

9.620787 

.621142 

.621497 

.621852 

.622207 

.622561 

.622915 

.623269 

.623623 

.623976 

9.624330 

.624633 

.625036 

.625338 

.625741 

.626093 

.626445 

.626797 

.627149 

.627501 

.627852 

0.393590 

.393227 

.392363 

.392500 

.392137 

.391775 

.391412 

.391050 

.390688 

.390326 

0.339964 

.389603 

.389241 

.383880 

.388520 

.338159 

.337799 

.387439 

.387079 

.386719 

0.336359 

.386000 

.385641 

.335282 

.334923 

.384565 

.384207 

.383849 

.333491 

.333133 

0.382776 

.332418 

.332061 

.381705 

.381348 

.330992 

.330636 

.330280 

.379924 

.379568 

0.379213 

.378358 

.378503 

.378148 

.377793 

.377439 

.377085 

.376731 

.376377 

.376024 

0.375670 

.375317 

.374964 

.374612 

.374259 

.373907 

.373555 

.373203 

.372851 

.372499 

.372148 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

23 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


1 13^ 


G7 a 











































23° 


COSINES, TANGENTS, AND COTANGENTS 


197 

15G 3 


M. 

Sine. 

D. 1”. 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Co tang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 
33 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
4S 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.591878 

.592176 

.592473 

.592770 

.593067 

.593363 

.593659 

.593955 

.594251 

.594547 

9.594842 

.595137 

.595432 

.595727 

.596021 

.596315 

.596609 

.596903 

.597196 

.597490 

9.597783 

.593075 

.598368 

.59S660 

.598952 

.599244 

.599536 

.599827 

.600118 

.600409 

9.600700 

.600990 

.601230 

.601570 

.601860 

.602150 

.602439 

.602728 

.603017 

.603305 

9.603594 

.603882 

.604170 

.604457 

.604745 

.605032 

.605319 

.605606 

.605892 

.606179 

9.606465 
.606751 
.607036 
.607322 
.607607 
.607892 
.608177 
.60'461 
.608745 
.609029 
.609313 

4.96 

4.95 

4.95 

4.95 

4.94 

4.94 

4.93 

4.93 

4.93 

4.92 

4.92 

4.91 

4.91 

4.91 

4.90 

4.90 

4.89 

4.89 

4.89 

4.88 

4.88 

4.88 

4.87 

4.87 

4.86 

4.86 

4.86 

4.85 

4.85 

4.84 

4.84 

4.84 

4.83 

4.83 

4.83 

4.82 

4.82 

4.81 

4.81 

4.81 

4. SO 
4.80 
4.79 
4.79 
4.79 
4.78 
4.78 
4.78 
4.77 
4.77 

4.76 

4.76 

4.76 

4.75 

4.75 

4.74 

4.74 

4.74 

4 73 
4.73 

9.964026 

.963972 

.963919 

.963865 

.963811 

.963757 

.963704 

.963650 

.963596 

.963542 

9.963488 

.963434 

.963379 

.96332? 

.963271 

'.§83217 

.963160 

.963108 

.963054 

.962999 

9.962945 

.962890 

.962836 

.962781 

.962727 

.962672 

.962617 

.962562 

.962503 

.962453 

9.962398 

.962343 

.962288 

.962233 

.962178 

.962123 

.962067 

.962012 

.961957 

.961902 

9.961846 

.961791 

.961735 

.961630 

.961624 

.961569 

.961513 

.961453 

.981402 

.961346 

9.961290 

.961235 

.961179 

.961123 

.961067 

.961011 

.960955 

.960899 

.960843 

.960786 

.960730 

.89 

.89 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.9(2 

.92 

.92 

.92 

.92 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.94 

.94 

.94 

9.627852 

.628203 

.628554 

.628905 

.629255 

.629606 

.629956 

.630306 

.630656 

.631005 

9.631355 

.631704 

.632053 

.632102 

.632750 

.633099 

.633447 

.633795 

.634143 

.634490 

9.634838 

.635185 

.635532 

.635879 

.636226 

.636572 

.636919 

.637265 

.637611 

.637956 

9.638302 

.638647 

.638992 

.639337 

.639682 

.640027 

.640371 

.640716 

.641060 

.641404 

9.641747 

.642091 

.612434 

.642777 

.643120 

.643463 

.643806 

.644148 

.644490 

.644832 

9.645174 

.645516 

.645857 

.646199 

.616540 

.616881 

.647222 

.647562 

.647903 

.648243 

.648583 

5.85 

5.85 

5.85 

5.84 

5.84 

5.84 

5.83 

5.83 

5.83 

5.82 

5.82 

5.82 

5.81 

5.81 

5.81 

5.80 

5.80 

5.80 

5.79 

5.79 

5.79 

5.78 

5.78 

5.78 

5.78 

5.77 

5.77 

5.77 

5.76 

5.76 

5.76 

5.75 

5.75 

5.75 

5.74 

5.74 

5.74 

5.73 

5.73 

5.73 

5.73 

5.72 

5.72 

5.72 

5.71 

5.71 

5.71 

5.70 

5.70 

5.70 

5.69 

5.69 

5.69 

5.69 

5.68 

5.68 

5.68 

5.67 

5.67 

5.67 

0.372148 

.371797 

.371446 

.371095 

.370745 

.370394 

.370044 

.369694 

.369344 

.368995 

0.368645 

.368296 

.367947 

.367598 

.367250 

.366901 

.366553 

.366205 

.365857 

.365510 

0.365162 

.364315 

.364468 

.364121 

.363774 

.363428 

.3630S1 

.362735 

.362389 

.362044 

0.361698 

.361353 

.361008 

.360663 

.360318 

.359973 

.359629 

.359284 

.358940 

.358596 

0.358253 

.357909 

.357566 

.357223 

.356880 

.356537 

.356194 

.355852 

.355510 

.355168 

0.354826 

.354484 

.354143 

.353801 

.353460 

.353119 

.352778 

.352438 

.352097 

.351757 

.351417 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


GfiO 


1130 

















































198 

340 


TABLE XIII. LOGARITHMIC SINES, 

. 155« 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1'. 

t 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 
17 ■ 
13 

19 

20 

21 

22 

23 

24 

25 

26 
27 
23 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 
51 

55 

56 

57 
53 

59 

60 

9.609313 

.609597 

.609380 

.610164 

.610447 

.610729 

.611012 

.611294 

.611576 

.611853 

9.612140 

.612421 

.612702 

.612933 

.613264 

.613545 

.613325 

.614105 

.614335 

.614665 

9.614944 

.615223 

.615502 

.615781 

.616060 

.616333 

.616616 

.616394 

.617172 

.617450 

9.617727 

.618004 

.618231 

.618558 

.618834 

.619110 

.619386 

.619662 

.619938 

.620213 

9.620488 
.620763 
.621033 
.621313 
.6215S7 
.621861 
.622135 
.622409 
.622632 
. 622956 

9.623229 

.623502 

.623774 

.621047 

.624319 

.621591 

.624863 

.625135 

.625406 

.625677 

.625943 

4.73 

4.72 

4.72 

4.72 

4.71 

4.71 

4.71 

4.70 

4.70 

4.69 

4.69 

4.69 

4.68 

4.63 
4.68 
4.67 
4.67 
4.67 
4.66 
4.66 

4.65 

4.65 

4.65 

4.64 

4.61 
4.64 
4.63 
4.63 
4.63 

4.62 

4.62 

4.61 

4.61 

4.61 

4.60 

4.60 

4.60 

4.59 

4.59 

4.59 

4.53 

4.53 
4.58 
4.57 
4.57 
4.57 
4.56 
4.56 
4.56 
4.55 

4.55 

4.54 
4.54 
4.54 
4.53 
4.53 
4.53 
4.52 
4.52 
4.52 

9.960730 

.960674 

.960618 

.960561 

.960505 

.960448 

.960392 

.960335 

.960279 

.960222 

9.960165 

.960109 

.960052 

.959995 

.959933 

.959882 

.959325 

.959763 

.959711 

.959654 

9 959596 
.959539 
.959482 
.959425 
.959363 
.959310 
.959253 
.959195 
.959133 
.959030 

9.959023 

.958965 

.953903 

.958850 

.958792 

.958734 

.958677 

.958619 

.953561 

.958503 

9.958445 

.958387 

.958329 

.958271 

.958213 

.958154 

.958096 

.958033 

.957979 

.957921 

9.957863 

.957804 

.957746 

.957687 

.957628 

.957570 

.957511 

.957452 

.957393 

.957335 

.957276 

.91 

.94 

.94 

.94 

.94 

.94 

.94 

.94 

.94 

.94 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.98 

.93 

.93 

.93 

.93 

.98 

.93 

.93 

.93 

9.64S5S3 

.648923 

.649263 

.649602 

.649942 

.650231 

.650620 

.650959 

.651297 

.651636 

9.651974 
.652312 
.652650 
.652983 
.653326 
.653663 
.654000 
.654337 
.654674 
.65501l 

9.035348 

.655634 

.656020 

.656356 

.656692 

.657028 

.657364 

.657699 

.658034 

.658369 

9.658704 

.659039 

.659373 

.659708 

.660042 

.660376 

.660710 

.661043 

.661377 

.661710 

9.662043 

.662376 

.662709 

.663042 

.663375 

.663707 

.664039 

.664371 

.664703 

.665035 

9.665366 

.665698 

.666029 

.666360 

.666691 

.667021 

.667352 

.667632 

.663013 

.668343 

.668673 

5.67 

5.66 

5.66 

5.66 

5.65 

5.65 

5.65 

5.64 

5.64 

5.64 

5.64 

5.63 

5.63 

5.63 

5.62 

5.62 

5.62 

5.62 

5.61 

5.61 

5.61 

5.61 

5.60 

5.60 

5.60 

5.59 

5.59 

5.59 

5.53 

5.53 

5.58 

5.58 

5.57 

5.57 

5.57 

5.56 

5.56 

5.56 

5.56 

5.55 

5.55 

5.55 

5.54 
5.54 
5.54 
5.54 
5.53 
5.53 
5.53 
5.53 

5.52 

5.52 

5.52 

5.51 

5.51 

5.51 

5.51 

5.50 

5.50 

5.50 

0.351417 

.351077 

.350737 

.350398 

.350058 

.349719 

.349380 

.349041 

.348703 

.343364 

0.348026 

.347638 

.347350 

.347012 

.346674 

.346337 

.346000 

.345663 

.345326 

.344989 

0.344652 

.344316 

.343930 

.343644 

.343303 

.342972 

.342636 

.342301 

.341966 

.341631 

0.341296 

.340961 

.340627 

.340292 

.339958 

.339624 

.339290 

.338957 

.333623 

.338290 

0.337957 

.337624 

.337291 

.336958 

.336625 

.336293 

.335961 

.335629 

.335297 

.334965 

0.334634 

.331302 

.333971 

.333640 

.333309 

.332979 

.332648 

.332318 

.331987 

.331657 

.331327 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


OS ' 1 


1140 







































35° 


COSINES, TANGENTS, AND COTANGENTS 


199 

1540 


M. 

Sine. 

D. 1". 

Cosine. 

D 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 
31 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 
59' 
60 

9.625948 

.626219 

.626490 

.626760 

.627030 

.627300 

.627570 

.627840 

.628109 

.628378 

9.628647 

.628916 

.629185 

.629453 

.629721 

.629989 

.630257 

.630524 

.630792 

.631059 

9.631326 

.631593 

.631859 

.632125 

.632392 

.632658 

.632923 

.633189 

.633454 

.633719 

9.633984 

.634249 

.634514 

.634778 

.635042 

.635306 

.635570 

.635834 

.636097 

.636360 

9.636623 

.636886 

.637148 

.637411 

.637673 

.637935 

.638197 

.638458 

.638720 

.638981 

9.639242 

.639503 

.639764 

.640024 

.640284 

.640544 

.640804 

.641064 

.641324 

.641583 

.641842 

4.51 

4.51 

4.51 

4.50 

4.50 

4.50 

4.49 

4.49 

4.49 

4.48 

4.48 

4.48 

4.47 

4.47 

4.47 

4.46 

4.46 

4.46 

4.45 

4.45 

. 4.45 
4.44 
4.44 
4.44 
4.43 
4.43 
4.43 
4.42 
4.42 
4.42 

4.41 

4.41 

4.41 

4.40 

4.40 

4.40 

4.39 

4.39 

4.39 

4.33 

4.38 

4.33 
4.37 
4.37 
4.37 
4.36 
4.36 
4.36 
4.35 
4.35 

4.35 

4.34 
4.34 
4.34 
4.33 
4.33 

4 33 
4.32 
4.32 
4.32 

9.957276 

.957217 

.957158 

.957099 

.957040 

.956981 

.956921 

.956862 

.956803 

.956744 

9.956684 

.956625 

.956566 

.956506 

.956447 

.956387 

.956327 

.95626S 

.956208 

.956148 

9.956089 

.956029 

.955969 

.955909 

.955849 

.955789 

.955729 

.955669 

.955609 

.955548 

9.955488 

.955428 

.955368 

.955307 

.955247 

.955186 

.955126 

.955065 

.955005 

.954944 

9.954883 

.954823 

.954762 

.954701 

.954640 

.954579 

.954518 

.954457 

.954396 

.954335 

9.954274 

.954213 

.954152 

.954090 

.954029 

.953968 

.953906 

.953845 

.953783 

.953722 

.953660 

.98 

.98 

.98 

.98 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.03 

1.03 

1.03 

9.66S673 

.669002 

.669332 

.669661 

.669991 

.670320 

.670649 

.670977 

.671306 

.671635 

9.671963 

.672291 

.672619 

.672947 

.673274 

.673602 

.673929 

.674257 

.674584 

.674911 

9.675237 

.675564 

.675890 

.676217 

.676543 

.676869 

.677194 

.677520 

.677846 

.678171 

9.678496 

.678821 

.679146 

.679471 

.679795 

.680120 

.680444 

.680768 

.681092 

.681416 

9.681740 

.682063 

.682387 

.682710 

.683033 

.683356 

.683679 

.6S4001 

.684324 

.634646 

9.68496S 

.685290 

.685612 

.685934 

.686255 

.686577 

.686898 

.687219 

687540 

687861 

.688182 

5.50 

5.49 

5.49 

5.49 

5.49 

5.48 

5.48 

5.48 

5.47 

5.47 

5.47 

5.47 

5.46 

5.46 

5.46 

5.46 

5.45 

5.45 

5.45 

5.45 

5.44 

5.44 

5.44 

5.44 

5.43 

5.43 

5.43 

5.42 

5.42 

5.42 

5.42 

5.41 

5.41 

5.41 

5.41 

5.40 

5.40 

5.40 

5.40 

5.39 

5.39 

5.39 

5.39 

5.38 

5.38 

5.38 

5.38 

5.37 

5.37 

5.37 

5.37 

5.36 

5.36 

5.36 

5.36 

5.35 

5.35 

5.35 

5.35 

5.35 

0.331327 

.330998 

.330668 

.330339 

.330009 

.329680 

.329351 

.329023 

.328694 

.328365 

0.328037 

.327709 

.327381 

.327053 

.326726 

.326398- 

.326071 

.325743 

.325416 

.325089 

0.324763 

.324436 

.324110 

.323783 

.323457 

.323131 

.322806 

.322480 

.322154 

321829 

0.321504 

.321179 

.320854 

.320529 

.320205 

.319880 

.319556 

.319232 

.318908 

.318584 

0.318260 

.317937 

.317613 

.317290 

.316967 

.316644 

.316321 

.315999 

.315676 

.315354 

0.315032 
.314710 
.314388 
.314066 
.313745 
.313423 
- .313102 
.312781 
.312460 
.312139 
.311818 

60 

59 

58 

57 

66 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

38 
35 
34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 ' 
0 

M. | 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1'. 

Tang. 1 

M. 


1150 


64° 






































200 

ar»o 


TABLE XIII. LOGARITHMIC SINES, 

153* 


M. 

Siue. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 
82 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.641842 

.642101 

.642360 

.642618 

.642877 

.643135 

.643393 

.643650 

.643903 

.644165 

9.644123 

.644580 

.644936 

.645193 

.645450 

.645706 

.645962 

.646218 

.646474 

.646729 

9.646984 

.647240 

.647494 

.647749 

.648004 

.648253 

.648512 

.643766 

.649020 

.649274 

9.649527 

.649781 

.650034 

.650287 

.650539 

.650792 

.651044 

.651297 

.651549 

.651800 

9.652052 

.652304 

.652555 

.652806 

.653057 

.653303 

.653558 

.653803 

.654059 

.654309 

9.654558 

.654803 

.655058 

.655307 

.655556 

.655305 

.656054 

.656302 

.656551 

.656799 

.657047 

4.32 

4.31 

4.31 

4.31 

4.30 

4.30 

4.30 

4 29 
4.29 
4.29 

4.28 

4.28 

4.23 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
4.26 

4.25 

4.25 

4.25 

4.24 
4.24 
4.24 
4.23 
4.23 
4.23 
4.22 

4.22 

4.22 

4.22 

4.21 

4.21 

4.21 

4.20 

4.20 

4.20 

4.19 

4.19 

4.19 

4.18 

4.18 

4.18 

4.18 

4.17 

4.17 

4.17 

4.16 

4.16 

4.16 

4.15 

4.15 

4.15 

4.15 

4.14 

4.14 

4.14 

4.13 

9.953660 

.953599 

.953537 

.953475 

.953413 

.953352 

.953290 

.953228 

.953166 

.953104 

9.953042 

.952980 

.952918 

.952855 

.952793 

.952731 

.952669 

.952606 

.952544 

.952481 

9.952419 

.952356 

.952294 

.952231 

.952168 

.952106 

.952043 

.951930 

.951917 

.951854 

9.951791 

.951728 

.951665 

.951602 

.951539 

.951476 

.951412 

.951349 

.931286 

.951222 

9.951159 

.951096 

.951032 

.950963 

.950905 

.950341 

.950778 

.950714 

.950650 

.950586 

9.950522 

.950453 

.950394 

.950330 

.950266 

.950202 

.950133 

.950074 

.950010 

.949945 

.949881 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1 03 
1.03 
1.03 

1.03 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.00 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

9.68S1S2 

.688502 

.688S23 

.639143 

.689463 

.689783 

.690103 

.690423 

.690742 

.691062 

9.691381 

.691700 

.692019 

.692338 

.692656 

.692975 

.693293 

.693612 

.693930 

.694248 

9.694566 

.694333 

.695201 

.695518 

.695836 

.696153 

.696470 

.696787 

.697103 

.697420 

9.697736 

.698053 

.698369 

.693635 

.699001 

.699316 

.699632 

.699947 

.700263 

.700578 

9.700393 

.701208 

.701523 

.701837 

.702152 

.702466 

.702731 

.703095 

.703409 

.703722 

9.704036 

.701350 

.704663 

.704976 

'.705290 

.705603 

.705916 

.706228 

.706541 

.706854 

.707166 

5.34 

5.34 

5.34 

5.34 

5.33 

5.33 

5.33 

5.33 

5.32 

5.32 

5.32 

5.32 

5.31 

5.31 

5.31 

5.31 

5.30 

5.30 

5.30 

5.30 

5.29' 

5.29 

5.29 

5.29 

5.29 

5.23 

5.28 

5.28 

5.23 
5.27 

5.27 

5.27 

5.27 

5.26 

5.26 

5.26 

5.26 

5.26 

5.25 

5.25 

5.25 

5.25 

5.24 
5.24 
5.24 
5.24 
5.24 
5.23 
5.23 
5.23 

5.23 

5.22 

5.22 

5.22 

5.22 

5.22 

5.21 

5 21 
5.21 
5.21 

0.311818 

.311498 

.311177 

.310857 

.310537 

.310217 

.309897 

.309577 

-309258 

.308933 

0.30S619 

.308300 

.307981 

.307662 

.307344 

.307025 

.306707 

.306388 

.306070 

.305752 

0.305434 

.305117 

.304799 

.304482 

.304164 

.303847 

303530 

.303213 

.302897 

.302580 

0.302264 

.301947 

.301631 

.301315 

.300999 

.300684 

.300363 

.300053 

.299737 

.299422 

0.299107 

.298792 

.293477 

.298163 

.297848 

.297534 

.297219 

.296905 

.296591 

.296278 

0.295964 

.295650 

.295337 

.295Q24 

.294710 

.294397 

.294084 

.293772 

.293459 

.293146 

.292334 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 




116° 

































COSINES, TANGENTS, AND COTANGENTS. 201 

270 1523 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 
39 

10 
11 
12 

13 

14 

15 

16 
17 
13 

49 

50 

51 

52 

53 

54 

55 

56 
! 57 

58 

5£ 

GC 

j M. 

9.657047 

.657295 

.657542 

.657790 

.653037 

.658234 

.653531 

.053773 

659025 

.659271 

9.659517 

.659763 

.660009 

.660255 

.660501 

.660746 

.660991 

.661236 

.661481 

.661726 

9.661970 

.662214 

.662459 

.662703 

.662946 

.663190 

.663433 

.663677 

.663920 

.664163 

9.664406 

.664648 

.664891 

.665133 

.665375 

.665617 

.665859 

.666100 

.666342 

.666533 

9.666S24 

.667065 

.667305 

.667546 

.667786 

.663027 

.663267 

.663506 

.663746 

.663936 

9.669225 

.669464 

.669703 

.669942 

.670181 

.670419 

.670653 

.670396 

.671134 

.67-1372 

.671609 

4.13 

4.13 

4.12 

4.12 

4.12 

4.12 

4.11 

4.11 

4.11 

4.10 

4.10 

4.10 

4.10 

4.09 

4.09 

4.09 

4.08 

4.08 

4.08 

4.08 

4.07 

4.07 

4.07 

4.06 

4.06 

4.06 

4.05 

4.05 

4.05 

4.05 

4.04 

4.04 

4.04 

4.03 

4.03 

4.03 

4.03 

4.02 

4.02 

4.02 

4.01 

4.01 

4.01 

4.01 

4.00 

4.00 

4.00 

3.99 

3.99 

3.99 

3.99 

3.93 

3.98 

3.98 

3.93 

3.97 

3.97 

3.97 

3.96 

3.96 

9.949331 

.919816 

.949752 

.949638 

.949623 

.949553 

.949494 

.949429 

.949364 

.949300 

9.949235 

.949170 

.949105 

.949040 

.948975 

.948910 

.948345 

.943730 

.943715 

.948650 

9.948584 

.948519 

.943454 

.948338 

.943323 

.948257 

.948192 

.943126 

.943060 

.917995 

9.947929 

.947863 

.947797 

.947731 

.947665 

.947600 

.947533 

.947467 

.947401 

.947335 

9.947269 

.947203 

.947136 

.947070 

.947004 

.946937 

.946371 

.946304 

.946738 

.946671 

9.946604 

.946533 

.946471 

.946404 

.946337 

.946270 

.946203 

.946136 

.946069 

.946002 

.945935 

1.07 

1.07 

1.07 

1.08 

1.03 

1.08 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.08 

1.03 

1.03 

1.08 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.09 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.12 

1.12 

1.12 

1.12 

1.12 

1 12 

9.707166 

.707478 

.707790 

.708102 

.708414 

.703726 

.709037 

.709349 

.709660 

.709971 

9.710282 

.710593 

.710904 

.711215 

.711525 

.711836 

.712146 

.712456 

.712766 

.713076 

9.713386 

.713696 

.714005 

.714314 

.714624 

.714933 

.715242 

.715551 

.715860 

.716163 

9.716477 

.716785 

.717093 

.717401 

.717709 

.718017 

.718325 

.718633 

.718940 

.719248 

9.719555 

.719362 

.720169 

.720476 

.720783 

.721089 

.721396 

.721702 

.722009 

.722315 

9.722621 

.722927 

.723232 

.723538 

.723844 

.724149 

.724454 

.724760 

.725065 

.725370 

.725674 

5.20 

5.20 

5.20 

5.20 

5.20 

5.19 

5.19 

5.19 

5.19 

5.18 

5.18 

5.18 

5.18 

5.18 

5.17 

5.17 

5.17 

5.17 

5.17 

5.16 

5.16 

5.16 

5.16 

5.15 

5.15 

5.15 

5.15 

5.15 

5.14 

5.14 

5.14 

5.14 

5.14 

5.13 

5.13 

5.13 

5.13 

5.13 

5.12 

5.12 

5.12 

5.12 

5.11 

5.11 

5.11 

5.11 

5.11 

5.10 

5.10 

5.10 

5.10 

5.10 

5.09 

5.09 

5.09 

5.09 

5.09 

5.03 

5.08 

5.03 

0.292834 

.292522 

.292210 

.291898 

.291586 

.291274 

.290963 

.290651 

.290340 

.290029 

0.289718 

.289407 

.289096 

.238785 

.288475 

.288164 

.287854 

.237544 

.287234 

.236924 

0.286614 

.286304 

.285995 

.285636 

.285376 

.285067 

.284758 

.284449 

.234140 

.283832 

0.283523 

.283215 

.282907 

.282599 

.282291 

.281983 

.281675 

.281367 

.231060 

.230752 

0.280445 

.280138 

.279831 

.279524 

.279217 

.278911 

.278604 

.278293 

.277991 

.277685 

0.277379 

.277073 

.276768 

.276462 

.276156 

.275851 

.275546 

.275240 

.274935 

.274630 

.274326 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

23 

27 

26 

25 

24 

23 

22 

21 

20 

19 

15 

17 

16 

15 

14 

13 

12 

11 

10 

9 . 
8 

7 

6 

5 

4 

3 

2 

1 

0 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

B.l". 1 Tang. 

M. 

1 



62° 


1 XT' 0 


10 












































202 TABLE XIII. LOGARITHMIC SINES, 

S8° 151 


M. 

Sine. 

D. 1". 1 Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

53 

59 

9.671609 

.671847 

.672084 

.672321 

.672558 

.672795 

.673032 

.673268 

.673505 

.673741 

9.673977 
.674213 
.674448 
.674684 
.674919 
.675155 
.675390 
.675624 
.675859 
.676094 

9.676328 

.676562 

.676796 

.677030 

.677254 

.677498 

.677731 

.677964 

.678197 

.678430 

9.678663 

.678895 

.679128 

.679360 

.679592 

.679824 

.630056 

.630233 

.630519 

.630750 

9.680982 

.631213 

.631443 

.631674 

.63,1905 

.632135 

.632365 

.682595 

.682325 

.683055 

9.633234 

.633514 

.633743 

.633972 

.634201 

.634430 

.634658 

.634887 

.635115 

.635343 

.635571 

3.96 

3.96 

3.95 

3.95 

3.95 

3.94 

3.94 

3.94 

3.94 

3.93 

3.93 

3.93 

3.93 

3.92 

3.92 

3.92 

3.91 

3.91 

3.91 

3.91 

3.90 

3.90 

3.90 

3.90 

3.89 

3.89 

3.89 

3.83 
3.88 
3.88 

3.88 

3.87 

3.87 

3.87 

3.87 

3.86 

3.86 

3.86 

3.86 

3.85 

3.85 

3.85 

3.84 
3.84 
3.84* 
3.84 
3.83 
3.83 
3.83 
3.83 

3.82 

3.82 

3.82 

3.82 

3.81 

3.81 

3.81 

3.80 

3.80 

3.80 

9.945935 

.945868 

.945800 

.945733 

.945666 

.945598 

.945531 

.945464 

.945396 

.945328 

9.945261 

.945193 

.945125 

.945058 

.944990 

.944922 

.941854 

.944786 

.944718 

.944650 

9.944582 

.944514 

.944446 

.944377 

.944309 

.944241 

.944172 

.944104 

.944036 

.943967 

9.943899 

.943830 

.943761 

.943693 

.943624 

.943555 

.943486 

.943417 

.943348 

.943279 

9.943210 

.943141 

.943072 

.94.3003 

.942934 

.942864 

.942795 

.942726 

.942656 

.942587 

9.942517 

.942448 

.942378 

.942308 

.942239 

.942169 

.942099 

.942029 

.941959 

.941889 

.941819 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 

1.14 

1.14 

1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 

1.15 

1.15 

1.15 

1.15 

1.15 

1.16 
1.16 
1.16 
1.16 
1.16 

1.16 

1.16 

1.16 

1.16 

1.16 

1.16 

1.16 

1.17 

1.17 

1.17 

9.725674 

.725979 

.726284 

.726588 

.726892 

.727197 

.727501 

.727805 

.728109 

.728412 

9.728716 

.729020 

.729323 

.729626 

.729929 

.730233 

.730535 

.730338 

.731141 

.731444 

9.731746 

.732048 

.732351 

.732653 

.732955 

.733257 

.733558 

.733860 

.734162 

.734463 

9.734764 

.735066 

.735367 

.735668 

.735969 

.736269 

.736570 

.736870 

.737171 

.737471 

9.737771 

.738071 

.738371 

.733671 

.738971 

.739271 

.739570 

.739870 

.740169 

.740468 

9.740767 

.741066 

.741365 

.741664 

.741962 

.742261 

.742559 

.742858 

.743156 

.743454 

.743752 

5.08 

5.08 

5.07 

5.07 

5.07 

5.07 

5.07 

5.06 

5.06 

5.06 

5.06 

5.06 

5.05 

5.05 

5.05 

5.05 

5.05 

5.05 

5.04 

5.04 

5.04 

5.04 

5.04 

5.03 

5.03 

5.03 

5.03 

5.03 

5.02 

5.02 

5.02 

5.02 

5.02 

5.01 

5.01 

5.01 

5.01 

5.01 

5.01 

5.00 

5.00 

5.00 

5.00 

5.00 

4.99 

4.99 

4.99 

4.99 

4.99 

4.98 

4.98 

4.98 

4.93 

4.93 

4.98 

4.97 

4.97 

4.97 

4.97 

4.97 

0.274326 

.274021 

.273716 

.273412 

.273108 

.272803 

.272499 

.272195 

.271891 

.271588 

0.271284 

.270980 

.270677 

.270374 

.270071 

.269767 

.269465 

.269162 

.268859 

.26S556 

0.268254 

.267952 

.267649 

.267347 

.267045 

.266743 

.266442 

.266140 

.265838 

.265537 

0.265236 

.264934 

.264633 

.264332 

.264031 

.263731 

.263430 

.263130 

.262829 

.262529 

0.262229 

.261929 

.261629 

.261329 

.261029 

.260729 

.260430 

.260130 

.259331 

.259532 

0.259233 

.258934 

.258635 

.258336 

.258038 

.257739 

.257441 

.257142 

.256844 

.256546 

.256248 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

33 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

1 M. | 

1 — - 

Cosine, 

D. 1". 

Sine. 1 

D. 1". 

Cotang. | D. I". 

Tang. 



I 3 S' 


019 






























































29° 


COSINES, TANGENTS, AND COTANGENTS. 203 


1500 


M 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

■1 

i 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.635571 

.635799 

.686027 

.636254 

.636432 

.686709 

.636936 

.637163 

.637339 

.637616 

9.637843 

.633069 

.638295 

.633521 

.638747 

.633972 

.639193 

.639423 

.639648 

.639373 

9.690098 

.890323 

.690543 

.690772 

.690996 

.691220 

.691444 

.691663 

.691892 

.692115 

9.692339 

.692562 

.692735 

.693003 

.693231 

.693453 

.693676 

.693893 

.694120 

.694342 

9.694564 

.694786 

.695007 

.695229 

.695450 

.695671 

.695392 

.696113 

.696334 

.696554 

9.696775 

.696995 

.697215 

.697435 

.697654 

.697874 

.693094 

.698313 

.698532 

.693751 

.698970 

3.80 

3.79 

3.79 

3.79 

3.79 

3.78 

3.78 

3.78 

3.78 

3.77 

3.77 

3.77 

3.77 

3.76 

3.76 

3.76 

3.76 

3.75 

3.75 

3.75 

3.75 

3.74 

3.74 

3.74 

3.74 

3.73 

3.73 

3.73 

3.73 

3.72 

3.72 

3.72 

3.72 

3.71 

3.71 

3.71 

3.71 

3.70 

3.70 

3.70 

3.70 

3.69 

3.69 

3.69 

3.69 

3.63 

3.63 

3.63 

3.63 

3.67 

3.67 

3.67 

3.67 

3.66 

3.66 

3.66 

3.66 

3.65 

3.65 

3.65 

9.941819 

.941749 

.941679 

.941609 

.941539 

.941469 

.941393 

.941328 

.941258 

.941187 

9.941117 

.941046 

.940975 

.940905 

.940834 

.940763 

.940693 

.940622 

.940551 

.9404S0 

9.940409 

.940333 

.940267 

.940196 

.940125 

.940054 

.939982 

.939911 

.939840 

.939768 

9.939697 

.939625 

.939554 

.939482 

.939410 

.939339 

.939267 

.939195 

.939123 

.939052 

9.933980 

.93S903 

.933336 

.938763 

.933691 

.933619 

.933547 

.933475 

.933402 

.933330 

9.933258 
.933185 
.938113 
.933040 
.937967 • 
.937895 
.937822 
.937749 
.937676 
.93760-4 
.937531 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.13 
1.18 
1.13 
1.18 

1.18 

1.18 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.20 
1.20 
1.20 
1.20 
1.20 

1.20 

1.20 

1.20 

1.20 

I 20 
1.20 
1.20 
1.21 
1.21 
1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.22 

9.743752 

.744050 

.744343 

.744645 

.744943 

.745240 

.745538 

.745S35 

.746132 

.746429 

£. 146726 
.747023 
.747319 
.747616 
.747913 
.748209 
.743505 
.748801 
.749097 
.749393 

9.749639 
. 7499S5 
.750281 
.750576 
.750372 
.751167 
.751462 
.751757 
.752052 
.752347 

9.752642 

.752937 

.753231 

.753526 

.753320 

.754115 

.754409 

.754703 

.754997 

.755291 

9.755585 

.755378 

.756172 

.756165 

.756759 

.757052 

.757345 

.757633 

.757931 

.75S224 

9.753517 

.758810 

.759102 

.759395 

.759687 

.759979 

.760272 

.760564 

.760856 

.761148 

.761439 

4.96 

4.96 

4.96 

4.96 

4.96 

4.96 

4.95 

4.95 

4.95 

4.95 

4.95 

4.95 

4.94 

4.94 

4.94 

4.94 

4.94 

4.93 

4.93 

4.93 

4.93 

4.93 

4.93 

4.92 

4.92 

4.92 

4.92 

4.92 

4.92 

4.91 

4.91 

4.91 

4.91 

4.91 

4.91 

4.90 

4.90 

4.90 

4.90 

4.90 

4.89 

4.89 

4.89 

4.89 

4.89 

4.89 

4.83 

4.88 

4.83 

4.88 

4.88 

4.88 

4.87 

4.87 

4.87 

4.87 

4,37 

4.87 

4.86 

4.86 

0.256248 

.255950 

.255652 

.255355 

.255057 

.254760 

.254462 

.254165 

.253863 

.253571 

0.253274 

.252977 

.252681 

.2523S4 

.252087 

.251791 

.251495 

.251199 

.250903 

.250607 

0.250311 

.250015 

.249719 

.249424 

.249128 

.248833 

.248538 

.248243 

.247948 

.247653 

0.247358 

.247063 

.246769 

.246474 

.246180 

.245885 

.245591 

.245297 

.245003 

.244709 

0.244415 

.244122 

.243828 

.243535 

.243241 

.242948 

.242655 

.242362 

.242069 

.241776 

0.241433 

.241190 

.240898 

.240605 

.240313 

.240021 

.239728 

.239436 

.239144 

.233352 

.233561 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

23 
27 
26 
25 

24 
23 
22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


M. 

Cosine. 

D. 1». 

Sine. 

D. 1". 

Cotang. 1 

D. 1". 

Tang. 

M. 



1190 


GO* 



















































149° 


204 TABLE XIII. LOGARITHMIC SINES, 


30° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 
'26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.698970 

.699189 

.699407 

.699626 

.699844 

.700062 

.700280 

.700498 

.700716 

.700933 

9.701151 

.701368 

.701585 

.701802 

.702019 

.702236 

.702452 

.702669 

.702885 

.703101 

9.703317 

.703533 

.703749 

.703964 

.704179 

.704395 

.704610 

.704825 

.705040 

.705254 

9.705469 

.705683 

.705898 

.706112 

.706326 

.706539 

.706753 

.706967 

.707180 

.707393 

9.707606 

.707819 

.708032 

.708245 

.708458 

.708670 

.708882 

.709094 

.709306 

.709518 

9.709730 

.709941 

.710153 

.710364 

.710575 

.710786 

.710997 

.711208 

.711419 

.711629 

.711839 

| 

3.65 

3.64 

3.64 

3.64 

3.64 

3.63 

3.63 

3.63 

3.63 

3.62 

3.62 

3.62 

3.62 

3.61 

3.61 

3.61 

3.61 

3.60 

3.60 

3.60 

3.60 

3.59 

3.59 

3.59 

3.59 

3.59 

3.58 

3.58 

3.58 

3.58 

3.57 

3.57 

3.57 

3.57 

3.56 

3.56 

3.56 

3.56 

3.55 

3.55 

3.55 

3.55 

3.54 

3.54 

3.54 

3.54 

3.54 

3.53 

3.53 

3.53 

3.53 

3.52 

3.52 

3.52 

3.52 

3.51 

3.51 

3.51 

3.51 

3.51 

9.937531 
.937458 
' .937385 
.937312 
.937238 
.937165 
.937092 
.937019 
.936946 
.936872 

9.936799 

.936725 

.936652 

.936578 

.936505 

.936431 

.936357 

.936284 

.936210 

.936136 

9.936062 

.935988 

.935914 

.935840 

.935766 

.935692 

.935618 

.935543 

.935469 

.935395 

9.935320 

.935246 

.935171 

.935097 

.935022 

.934948 

.934873 

.934798 

.934723 

.934649 

9.934574 

.934499 

.934424 

.934349 

.934274 

.934199 

.934123 

.934048 

.933973 

.933398 

9.933822 

.933747 

.933671 

.933596 

.933520 

933445 

933369 

933293 

933217 

.933141 

.933066 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.24 
1.24 
1.24 
1.24 
1.24 
1.24 

1.24 

1.24 

1.24 

1.24 

1.24 

1.24 

1.25 
1.25 
1.25 
1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1 .26 
1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

9.761439 

.761731 

.762023 

.762314 

.762606 

.762S97 

.763188 

.763479 

.763770 

.764061 

9.764352 

.764643 

.764933 

.765224 

.765514 

.765805 

.766095 

.766385 

.766675 

.766965 

9.767255 

.767545 

.767834 

.768124 

.768414 

.768703 

.768992 

.769281 

.769571 

.769860 

9.770148 

.770437 

.770726 

.771015 

.771303 

.771592 

.771880 

.772168 

.772457 

.772745 

9.773033 

.773321 

.773608 

.773896 

.774184 

.774471 

.774759 

.775046 

.775333 

.775621 

9.775908 

.776195 

.776482 

.776768 

.777055 

.777342 

.777628 

.777915 

.778201 

.778488 

.778774 

4.86 

4.86 

4.86 

4.86 

4.86 

4.85 

4.85 

4.85 

4.85 

4.85 

4.85 

4.84 

4.84 

4.84 

4.84 

4.84 

4.84 

4.S3 
4.83 
4.83 

4.83 

4.83 

4.83 

4.82 

4.32 

4.82 

4.82 

4.S2 

4.82 

4.82 

4.81 

4.81 

4.81 

4.81 

4.81 

4.81 

4.80 

4.80 

4.80 

4.80 

4.80 

4.80 

4.80 

4.79 

4.79 

4.79 

4.79 

4.79 

4.79 

4.78 

4.78 

4.78 

4.78 

4.78 

4.78 

4.78 

4.77 

4.77 

4.77 

4.77 

0.238561 

.238269 

.237977 

.237686 

.237394 

.237103 

.236812 

.236521 

.236230 

.235939 

0.235648 

.235357 

.235067 

.234776 

.234486 

.234195 

.233905 

.233615 

.233325 

.233035 

0.232745 

.232455 

.232166 

.2.31876 

.231586 

.231297 

.231008 

.230719 

.230429 

.230140 

0.229852 

.229563 

.229274 

.228985 

.228697 

.228408 

.228120 

.227832 

.227543 

.227255 

0.226967 

.226679 

.226392 

.226104 

.225816 

.225529 

.225241 

.224954 

.224667 

.224379 

0.224092 

.223805 

.223518 

.223232 

.222945 

.222653 

.222372 

.222085 

.221799 

.221512 

.221226 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 : 
28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

I). 1". 

Tang. | 

M. 




120° 








































































COSINES, TANGENTS, AND COTANGENTS 


2Q£ 


31° 148 3 


M. 

Sine. 

D. 1". 

Cosine. 1 

D. 1". ! 

Tang. 

D. 1". 

Cotang 

M. 

60 

59 | 
58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 
23 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

M. 

9.711839 

.712050 

.712260 

.712469 

.712679 

.712839 

.713098 

.713303 

.713517 

.713726 

9.713935 

.714144 

.714352 

.714561 

.714769 

.714973 

.715136 

.715394 

.715602 

.715809 

9.716017 

.716224 

.716432 

.716639 

.716846 

.717053 

.717259 

.717466 

.717673 

.717879 

9.718085 

.718291 

.718497 

.718703 

.718909 

.719114 

.719320 

.719525 

.719730 

.719935 

9.720140 

.720345 

.720549 

.720754 

.720958 

.721162 

.721366 

.721570 

.721774 

.721978 

9.722181 

.722385 

.722588 

.722791 

.722994 

.723197 

.723400 

.723603 

.723305 

.724007 

.724210 

3.50 

3.50 

3.50 

3.50 

3.49 

3.49 

3.49 

3.49 

3.48 

3.48 

3.48 

3.43 

3.43 
3.47 
3.47 
3.47 
3.47 
3.46 
3.46 
3.46 

3.46 

3.46 

3.45 

3.45 

3.45 

3.45 

3.44 
3.44 
3.44 
3.44 

3.43 

3.43 

3.43 

3.43 

3.43 

3.42 

3.42 

3.42 

3 42 
3.41 

3.41 

3.41 

3.41 

3.41 

3.40 

3.40 

3.40 

3.40 

3.39 

3.39 

3.39 

3.39 

3.39 

3.38 

3.38 

3.33 
3.33 
3.37 
3.37 
3.37 

9.933066 

.932990 

.932914 

.932333 

.932762 

.932635 

.932609 

.932533 

.932457 

.932380 

9.932304 

.932228 

.932151 

.932075 

.931998 

.931921 

.931845 

.931768 

.931691 

.931614 

9.931537 

.931460 

.931383 

.931306 

.931229 

.931152 

.931075 

.930993 

.930921 

.930343 

9.930766 

.930638 

.930611 

.930533 

.930456 

.930378 

.930300 

.930223 

.930145 

.930067 

9.929989 
.929911 
.929333 
.929755 
.929677 
.929599 
.929521 
.929442 
.929364 
.929286 

9.929207 

.929129 

.929050 

.928972 

.928893 

.923815 

.923736 

.923657 

.923578 

.928499 

.923420 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.28 
1.28 
1.28 
1.23 
1.23 
1.28 
1.28 
1.28 

1.28 

1.28 

1.23 

1.23 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1.30 
1.30 
1.30 
1.30 

1.30 

1.30 

1.30 

1.30 

1.30 

1.30 

1.30 

1.31 
1.31 
1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.32 

9.778774 

.779060 

.779346 

.779632 

.779918 

.780203 

.780489 

.780775 

.781060 

.781346 

9.781631 

.781916 

.782201 

.782486 

.782771 

.783056 

.783341 

.783626 

.783910 

.784195 

9 784479 
784764 
785048 
.785332 
.785616 
785900 
.786184 
.786463 
.786752 
.787036 

9.787319 

.787603 

.787886 

.788170 

.788453 

.788736 

.789019 

.789302 

.789535 

.789368 

9.790151 

.790434 

.790716 

.790999 

.791281 

.791563 

.791846 

.792123 

.792410 

.792692 

9.792974 

.793256 

.793538 

.793819 

.794101 

.794383 

.794664 

.794946 

.795227 

.795508 

.795789 

4.77 

4.77 

4.77 

4.76 

4.76 

4.76 

4.76 

4.76 

4.76 

4.76 

4.75 

4.75 

4.75 

4.75 

4.75 

4.75 

4.75 

4.74 

4.74 

4.74 

4.74 

4.74 

4.74 

4.74 

4.73 

4.73 

4.73 

4.73 

4.73 

4.73 

4.73 

4.72 

4.72 

4.72 

4.72 

4.72 

4.72 

4.72 

4.71 

4.71 

4.71 

4.71 

4.71 

4.71 

4.71 

4.70 

4.70 

4.70 

4.70 

4.70 

4.70 

4.70 

4.70 

4.69 

4.69 

4.69 

4.69 

4.69 

4.69 

4.69 

0.221226 

.220940 

.220654 

.220368 

.220032 

.219797 

.219511 

.219225 

.218940 

.218654 

0.218369 

218034 

.217799 

.217514 

.217229 

.216944 

.216659 

.216374 

.216090 

.215805 

0.215521 
.215236 
.214952 
.214663 
.214384 
.214100 
.213816 
.213532 
. .213248 
.212964 

0.212681 

.212397 

.212114 

.211830 

.211547 

211264 

.210981 

.210698 

.210415 

.210132 

0.209849 

.209566 

.209284 

.209001 

.208719 

.203437 

.208154 

.207872 

.207590 

.207308 

0.207026 

.206744 

.206162 

.206181 

.205899 

.205617 

.205336 

.205054 

.204773 

.204492 

.204211 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M 


iai’ 


5S a 








































TABLE XIII 


206 


LOGATIITIIMIC SINES, 


33° 147C 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

\ 57 

58 

59 

60 

9.724210 

.724412 

.724614 

.724816 

.725017 

.725219 

.725420 

.725622 

.725823 

.726024 

9.726225 

.726426 

.726626 

.726827 

.727027 

.727228 

.727428 

.727628 

.727828 

.728027 

9.728227 

.728427 

.728626 

.728825 

.729024 

.729223 

.729422 

.729621 

.729320 

.730018 

9.730217 

.730415 

.730613 

.730811 

.731009 

.731206 

.731404 

.731602 

.731799 

.731996 

9.732193 

.732390 

.732587 

.732784 

.732980 

.733177 

.733373 

.733569 

.733765 

.733961 

9.734157 

.734353 

.734549 

.734744 

.734939 

.735135 

.735330 

.735525 

.735719 

.735914 

.736109 

3.37 

3.37 

3.36 

3.36 

3.36 

3.36 

3.36 

3.35 

3.35 

3.35 

3.35 

3.34 

3.34 

3.34 

3.34 

3.34 

3.33 

3.33 

3.33 

3.33 

3.33 

3.32 

3.32 

3.32 

3.32 

3.31 

3.31 

3.31 

3.31 

3.31 

3.30 

3.30 

3.30 

3.30 

3.30 

3.29 

3.29 

3.29 

3.29 

3.28 

3.28 

3.28 

3.28 

3.28 

3.27 

3.27 

3.27 

3.27 

3.27 

3.26 

3.26 

3.26 

3.26 

3.26 

3.25 

3.25 

3.25 

3.25 

3.25 

3.24 

9.92S420 

.928342 

.928263 

.928183 

.928104 

.928025 

.927946 

.927667 

.927787 

.927708 

9.927629 

.927549 

".927470 

.927390 

.927310 

.927231 

.927151 

.927071 

.926991 

.926911 

9.926831 

.926751 

.926671 

.926591 

.926511 

.926431 

.926351 

.926270 

.926190 

.926110 

9.926029 

.925949 

.925868 

.925788 

.925707 

.925626 

.925545 

.925465 

.925384 

.925303 

9.925222 

.925141 

.925060 

.924979 

.924897 

.924816 

.924735 

.924654 

.924572 

.924491 

9.924409 

.924328 

.924246 

.924164 

.924083 

.924001 

.923919 

.923837 

.923755 

.923673 

.923391 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 

1.33 

1.33 

1.33 

1.34 
1.34 
1.34 
1.34 
1.34 
1.34 
1.34 

1.34 

1.34 

1.34 

1.34 

1.35 

1.35 
1.35 
1.35 
1.35 
1.35 

1.35 

1.35 

1.35 

1.35 

1.35 

1.35 

1.36 
1.36 
1.36 
1.36 

1.36 

1.36 

1.36 

1.36 

1.36 

1.36 

1.36 

1.37 
1.37 
1.37 

9.795789 

.796070 

.796351 

.796632 

.796913 

.797194 

.797474 

.797755 

.798036 

.798316 

9.798596 

.798877 

.799157 

.799437 

.799717 

.799997 

.800277 

.800557 

.800836 

.801116 

9.801396 

.801675 

.801955 

.802234 

.802513 

.802792 

.803072 

.803351 

.803630 

.803909 

9.804187 

.804466 

.804745 

.805023 

.805302 

.805580 

.805859 

.806137 

.806415 

.806693 

9.806971 

.807249 

.807527 

.807805 

.808083 

.808361 

.808638 

.808916 

.809193 

.809471 

9.809748 
.810025 
.810302 
.810580 
.810857 
.811134 
.811410 
.811687 
.811964 
.612241 
.812517 

4.68 

4.68 

4.68 

4.68 

4.68 

4.68 

4.68 

4.68 

4.67 

4.67 

4.67 

4.67 

4.67 

4.67 

4.67 

4t66 

4.66 

4.66 

4.66 

4.66 

4.66 

4.66 

4.66 

4.65 

4.65 

4.65 

4.65 

4.65 

4.65 

4 65 

4.65 

4.64 

4.64 

4.64 

4.64 

4.64 

4.64 

4.64 

4.64 

4.63 

4.63 

4.63 

4.63 

4.63 

4.63 

4.63 

4.63 

4.62 

4.62 

4.62 

4.62 

4.62 

4.62 

4.62 

4.62 

4.61 

4.61 

4.61 

4.61 

4.61 

0.204211 

.203930 

.203649 

.203363 

.203087 

.202806 

.202526 

.202245 

.201964 

.201684 

0.201404 

.201123 

.200843 

.200563 

.200283 

.200003 

.199723 

.199443 

.199164 

.198884 

0.198604 

.198325 

.198045 

.197766 

.197487 

.197208 

.196928 

.196649 

.196370 

.196091 

0.195813 

.195534 

.195255 

.194977 

.194698 

.194420 

.194141 

.193863 

.193585 

.193307 

0.193029 

.192751 

.192473 

.192195 

.191917 

.191639 

.191362 

.191084 

.190807 

.190529 

0.190252 

.189975 

.189698 

.189420 

.189143 

.188866 

.188590 

.188313 

.188036 

.187759 

.187483 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

38 

35 

34 

33 

32 

31 

30 

29 

28 

27 

28 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang 

I D. 1" 

Tang 

M. 


133 ° , 570 














































33 ° 


COSINES, TANGENTS, AND COTANGENTS 


20 ? 
1A G 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

d. r». 

Cotang. 

M. j 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 
33 

39 

40 

41 

42 
• 43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.736109 
.736303 
.736498 
. .736692 
.736886 
.737080 
.737274 
.737467 
.737661 
.737855 

9.733048 
.733241 
' .733434 
.738627 
.733820 
.739013 
.739206 
.739398 
.739590 
.739783 

9.739975 

.740167 

.740359 

.740550 

.740742 

.740934 

.741125 

.741316 

.741508 

.741699 

9.741889 

.742080 

.742271 

.742462 

.742652 

.742842 

.743033 

.743233 

.743413 

.743602 

9.743792 

.743932 

.744171 

.744361 

.744550 

.744739 

.744923 

.745117 

.745306 

.745594 

9.745683 

.745871 

.746060 

.746248 

.746136 

.746624 

.746312 

.746999 

.747187 

.747374 

.747562 

3.24 

3.24 

3.24 

3.23 

3.23 

3.23 

3.23 

3.23 

3.22 

3.22 

3.22 

3.22 

3.22 

3.21 

3.21 

3.21 

3.21 

3.21 

3.20 

3.20 

3.20 

3.20 

3.20 

3.19 

3.19 

3.19 

3.19 

3.19 

3.18 

3.18 

3.18 

3.18 

3.18 

3.17 

3.17 

3.17 

3.17 

3.17 

3.16 

3.16 

3.16 

3.16 

3.16 

3.15 

3.15 

3.15 

3.15 

3.15 

3.14 

3.14 

3.14 

3.14 

3.14 

3.13 

3.13 

3.13 

3.13 

3.13 

3.12 

3.12 

9.923591 

.923509 

.923427 

.923345 

.923263 

.923181 

.923098 

.923016 

.922933 

.922851 

9.922763 

.922686 

.922603 

.922520 

.922438 

.922355 

.922272 

.922189 

.922106 

.922023 

9.921940 

.921857 

.921774 

.921691 

.921607 

.921524 

.921441 

.921357 

.921274 

.921190 

9.921107 

.921023 

.920939 

.920356 

.920772 

.920688 

.920604 

.920520 

.920436 

.920352 

9.920268 

.920184 

.920099 

.920015 

.919931 

.919846 

.919762 

.919677 

.919593 

.919508 

9.919424 

.919339 

.919254 

.919169 

.919035 

.919000 

.918915 

.918830 

.918745 

.918659 

.918574 

1.37 

1.37 

1.37 

1.37 

1.37 

1.37 

1.37 

1.37 

1.37 
1.33 

1.38 
1.38 
1.38 
1.33 
1.38 
1.38 
1.38 
1.38. 
1.38 

1.38 

1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 

1.39 

1.39 

1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 

1.40 

1.40 

1.40 

1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 

1.41 

1.41 

1.41 

1.41 

1.42 
1.42 
1.42 
1.42 
1.42 
1.42 

9.812517 

.812794 

.813070 

.813347 

.813623 

.813899 

.814176 

.814452 

.814723 

.815004 

9.815280 

.815555 

.815831 

.816107 

.816382 

.816658 

.816933 

.817209 

.817484 

.817759 

9.818035 

.818310 

.818585 

.818860 

.819135 

.819410 

.819634 

.819959 

.820234 

.820508 

9.820783 

.821057 

.821332 

.821606 

.821880 

.822154 

.822429 

.822703 

.822977 

.823251 

9.823524 

.823798 

.824072 

.824345 

.824619 

.824893 

.825166 

.825439 

.825713 

.825986 

9.826259 

.826532 

.826305 

.827078 

.827351 

.827624 

.827897 

.828170 

.828442 

.823715 

.823987 

4.61 

4.61 

4.61 

4.61 

4.60 

4.60 

4.60 

4.60 

4.60 

4.60 

4.60 

4.60 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 

4.58 

4.58 

4.58 

4.58 

4.58 

4.58 

4.58 

4.58 

4.58 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.54 

4.54 

4.54 

0.187483 

.187206 

.186930 

.186653 

.186377 

.186101 

.185824 

.185548 

.185272 

.184996 

0.184720 

.184445 

.184169 

.183893 

.183618 

.183342 

.183067 

.182791 

.182516 

.182241 

0.181965 

.181690 

.181415 

.181140 

.180365 

.180590 

.180316 

.180041 

.179766 

.179492 

0.179217 
.178943 
.178668 
.178394 
.178120 
.177846 
.177571 
.177297 
.177023 
.176749 

0.176476 

.176202 

.175928 

.175655 

.175381 

.175107 

.174834 

.174561 

.174287 

.174014 

0.173741 
.173468 
.173195 
.172922 
.172649 
.172376 
.172103 
.171830 
. .171558 
.171285 
.171013 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

33 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

1 Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

1 Tang. 

M. 


56 a 


1330 



































208 TABLE XIII. LOGARITHMIC SINES, 


;j4° 145° 


M. 

Sine. 

». 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

JO 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 
•25 
26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.747562 

.747749 

.747936 

.748123 

.748310 

.748497 

.748683 

.743370 

.749056 

.749243 

9.749129 

.749615 

.749801 

.749987 

.750172 

.750358 

.750543 

.750729 

.750914 

.751099 

9.751284 

.751469 

.751654 

.751839 

.752023 

.752208 

.752392 

.752576 

.752760 

.752944 

9.753123 

*.753312 

.753495 

.753679 

.753862 

.754046 

.754229 

.754412 

.754595 

.754778 

9.754960 

.755143 

.755326 

.755503 

.755690 

.755872 

.756054 

.756236 

.756418 

.756600 

9.756782 

.756963 

.757144 

.757326 

.757507 

.757688 

.757869 

.758050 

.758230* 

.753411 

.75S591 

3.12 

3.12 

3.12 

3.11 

3.11 

3.11 

3.11 

3.11 

3.10 

3.10 

3.10 

3.10 

43.10 

3.10 
3.09 
3.09 
3.09 
3.09 
3.09 
3.08 

3.08 

3.08 

3.08 

3.08 

3.07 

3.07 

3.07 

3.07 

3.07 

3.06 

3.06 

3.06 

3.06 

3.06 

3.05 

3.05 

3.05 

3.05 

3.05 

3.05 

3.04 

3.04 

3.04 

3.04 

3.04 

3.03 

3.03 

3.03 

3.03 

3.03 

3.02 

3.02 

3.02 

3.02 

3.02 

3.02 

3.01 

3.01 

3.01 

3.01 

9.918574 

.918489 

.918404 

.918318 

.918233 

.918147 

.918062 

.917976 

.917891 

.917805 

9.917719 

.917634 

.917548 

.917462 

.917376 

.917290 

.917204 

.917118 

.917032 

.916946 

9.916859 

.916773 

.916637 

.916600 

.916514 

.916427 

.916341 

.916254 

.916167 

.916081 

9.915994 

.915907 

.915820 

.915733 

.915646 

.915559 

.915472 

.915385 

.915297 

.915210 

9.91512.3 

.915035 

.914948 

.914860 

.914773 

.914635 

.914598 

.914510 

.914422 

.914334 

9.914246 

.914158 

.914070 

.913982 

.913894 

.913866 

.913718 

913630 

.913541 

.913453 

.913365 

1.42 

1.42 

1.42 

1.42 

1.42 

1 43 

1.43 
1.43 
1.43 
1.43 

1.43 

1.43 

1.43 

1.43 

1.43 

1.43 

1.43 

1.44 
1.44 
1.44 

1.44 

1.44 

1.44 

1.44 

1.44 

1.44 

1.44 

1.44 

1.45 
1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 

9.823987 

.829260 

.829532 

.829805 

.830077 

.830349 

.830621 

.830893 

.831165 

.831437 

9.831709 

.831931 

.832253 
.832525 
.832796 
.833068 
.833339 
.833611 
.833382 
.834154 

9.834425 

.834696 

.834967 

.835233 

.835509 

.835780 

.836051 

.836322 

.836593 

.836864 

9.837134 

.837405 

.837675 

.837946 

.838216 

.838487 

.838757 

.839027 

.839297 

.839563 

9.839838 

.840108 

.840378 

.840648 

.840917 

.841187 

.841457 

.841727 

.841996 

.842266 

9.842535 

.842805 

.843074 

.843343 

.843612 

.843882 

.644151 

.844420 

.844689 

.844958 

.845227 

4.54 

4.54 

4.54 

4.54 

4.54 

4.54 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.52 

4.52 

4.52 

4.52 

4.52 

4.52 

4.52 

4.52 

4.52 

4.52 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.48 

4.48 

4.48 

4.43 

0.171013 

.170740 

.171*463 

.170195 

.169923 

.169651 

.169379 

.169107 

.163835 

.168563 

0.168291 

.16S019 

.167747 

.167475 

.167204 

.166932 

.166661 

.166389 

.166118 

.165846 

0.165575 

.165304 

.165033 

.164762 

.164491 

.164220. 

.163949 

.163678 

.163407 

.163136 

0.162866 

.162595 

.162325 

.162054 

.161784 

.161513 

.161243 

.160973 

.160703 

.160432 

0.160162 

.159892 

.159622 

.159352 

.159083 

.158813 

.158543 

.158273 

.158004 

.157734 

0.157465 

.157195 

.156926 

.156657 

.156388 

.156118 

.155849 

.155580 

.155311 

.154773 j 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

i 

*, 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 1 

M. 


533 




































350 


COSINES,, TANGENTS, AND COTANGENTS 


209 

1440 


M. 

Sine. 

D. 1" 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.758591 

.758772 

.758952 

.759132 

.759312 

.759492 

.759672 

.759852 

.760031 

.760211 

9.760390 

.760569 

.760748 

.760927 

.761106 

.761285 

.761464 

.761642 

.761821 

.761999 

9.7C2177 
.762356 
.762534 
.762712 
.762889 
.763067 
.763245 
.763422 
.763600 
.763777 

9.763954 

.764131 

.764308 

.764485 

.764662 

.764838 

.765015 

.765191 

.765367 

.765544 

9.765720 

.765896 

.766072 

.766247 

.766423 

.766598 

.766774 

.766949 

.767124 

.767300 

9.767475 

.767649 

.767824 

.767999 

.768173 

.768348 

.768522 

.768697 

.768871 

.769045 

.769219 

3.01 

3.00 

3.00 

3.00 

3.00 

3.00 

2.99 

2.99 

2.99 

2.99 

2.99 

2.99 

2.98 

2.98 

2.98 

2.98 

2.98 

2.97 

2.97 

2.97 

2.97 

2.97 

2.97 

2.96 

2.96 

2.96 

2.96 

2.96 

2.95 

2.95 

2.95 
2.95 
2.95 
2.95 
' 2.94 
2.94 
2.94 
2.94 
2.94 
2.93 

2.93 
2.93 
2.93 
2.93 
2.93 
2.92 
2.92 
• 2.92 
2.92 
2.92 

2.91 

2.91 

2.91 

2.91 

2.91 

2.91 

2.90 

2.90 

2.90 

2.90 

9,913365 

.913276 

.913187 

.913099 

.913010 

.912922 

.912833 

.912744 

.912655 

.912566 

9.912477 

.912388 

.912299 

.912210 

.912121 

.912031 

.911942 

.911853 

.911763 

.911674 

9.911584 

.911495 

.911405 

.911315 

.911226 

.911136 

.911046 

.910956 

.910866 

.910776 

9.910686 

.910596 

.910506 

.910415 

.910325 

.910235 

.910144 

.910054 

.909963 

.909873 

9.909782 

.909691 

.909601 

.909510 

.909419 

.909328 

.909237 

.909146 

.909055 

.908964 

9.908873 

.908781 

.908690 

.908599 

.908507 

.908416 

.908324 

.908233 

.908141 

.903049 

.907958 

1.47 

1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 

1.48 

1.48 

1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 

1.49 

1.49 

1.49 

1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 

1.50 

1.50 

1.50 

1.51 
1.51 
1.51 
1.51 
1.51 
1.51 
1.51 

1.51 

1.51- 

1.51 

1.51 

1.52 
1.52 
1.52 
1.52 
1.52 
1.52 

1.52 

1.52 

1.52 

1.52 

1.52 

1.53 
1.53 
1.53 
1.53 
1.53 

9.845227 

.845496 

.845764 

.846033 

.846302 

.846570 

.846839 

.847108 

.847376 

.847644 

9.847913 

.848181 

.848449 

.848717 

.848986 

.849254 

.849522 

.849790 

.850057 

.350325 

9.850593 

.850861 

.851129 

.851396 

.851664 

.851931 

.852199 

.852466 

.852733 

.853001 

9.853268 

.853535 

.853802 

.854069 

.854336 

.854603 

.854870 

.855137 

.855404 

.855671 

9.855933 

.856204 

.856471 

.856737 

.857004 

.857270 

.857537 

.857803 

.858069 

.858336 

9.858602 

.858868 

.859134 

.859400 

.859666 

.859932 

.860198 

.860464 

.860730 

.860995 

.861261 

4*48 

4.48 

4.48 

4.48 

4.48 

4.48 

4.48 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 ' 

4.43 

0.154773 

.154504 

.154236 

.153967 

.153698 

153430 

.153161 

.152892 

.152624 

.152356 

0.152087 

.151819 

.151551 

.151283 

.151014 

.150746 

.150478 

.150210 

.149943 

.149675 

0.149407 

.149139 

.148871 

.148604 

.148336 

.148069 

.147801 

.147534 

.147267 

.146999 

0.14 6732. 
.146465 
.146198 
.145931 
.145664 
.145397 
.145130 
.144863 
.144596 
.144329 

0.144062 

.143796 

.143529 

.143263 

.142996 

.142730 

.142463 

.142197 

.141931 

.141664 

0.141398 

.141132 

.140866 

.140600 

.140334 

.140068 

.139802 

.139536 

.139270 

.139005 

.138739 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

— 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. | 

M. 


1250 5 * 



















































‘210 

3G° 


TABLE XIII. LOGARITHMIC SINES, 

J 43* 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

9.769219 

2^90 

2.90 

2.89 

2.89 

2.89 

2.89 

2.89 

2.88 

2.88 

2.83 

9.907958 

1.53 

1.53 

1.53 

1.53 

1.53 

1.53 

1.54 
1.54 
1.54 
1.54 

9.861261 

4.43 

4.43 

4.43 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

0.138739 

60 

1 

.769393 

.907866 

.861527 

.133473 

59 

2 

.769566 

.907774 

.861792 

.133208 

58 

3 

.769740 

.907682 

.862058 

.137942 

57 

4 

5 

6 

.769913 

.770087 

.770260 

.907590 

.907498 

.907406 

.862323 

.862589 

.862354 

.137677 

.137411 

.137146 

56 

55 

54 

7 

.770433 

.907314 

.863119 

.136881 

53 

8 

9 

.770606 

.770779 

.907222 

.907129 

.863385 

.863650 

.136615 

.136350 

52 

51 

10 

9.770952 

2.88 

2.88 

2.88 

2.87 

2.87 

2.87 

2.87 

2.87 

2.87 

2.86 

9.907037 

1.54 

1.54 

1.54 

1.54 

1.54 

1.54 

1.55 
1.55 
1.55 
1.55 

9.863915 

4.42 
. 4.42 
4.42 
4.42 
4.42 
4.41 
4.41 
4.41 
4.41 
4.41 

0.136085 

50 

11 

.771125 

.906945 

.864180 

.135820 

49 

12 

.771293 

.906352 

.861445 

.135555 

48 

13 

.771470 

.906760 

.864710 

.135290 

47 

14 

.771813 

.906667 

.864975 

.135025 

46 

15 

.771815 

.906575 

.865240 

.134760 

45 

16 

.771987 

.906482 

.865505 

.134495 

44 

17' 

.772159 

.906389 

.865770 

.134230 

43 

18 

.772331 

.906296 

.866035 

.133965 

42 

19 

.772503 

.906204 

.866300 

.133700 

41 

20 

9.772675 

2.86 

2.S6 

2.86 

2.86 

2.85 

2.85 

2.85 

2.85 

2.85 

2.85 

9.906111 

1.55 

1.55 

1.55 

1.55 

1.55 

1.55 

1.55 

1.56 
1.56 
1.56 

9.866564 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.40 

4.40 

0.133436 

40 

21 

.772847 

.906018 

.866829 

.133171 

39 

22 

.773018 

.905925 

.867094 

.132906 

38 

23 

24 

.773190 

.773361 

.905832 

.905739 

.867358 

.867623 

.132642 

.132377 

37 

36 

25 

.773533 

.905645 

.867887 

.132113 

35 

26 

.773701 

.905552 

.863152 

.131848 

34 

27 

.773375 

.905459 

.S6S416 

.131584 

33 

28 

.774016 

.905366 

.863680 

.131320 

32 

29 

.774217 

.905272 

.863945 

.131055 

31 

30 

0.7743SS 

2.84 

2.84 

2.84 

2.84 

2.84 

2.84 

2.83 

2. S3 
2.83 
2.83 

9.905179 

1.56 

1.56 

1.56 

1.56 

1.56 

1.56 

1.56 

1.57 
1.57 
1.57 

9.869209 

4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
. 4.40 
4.40 

0.130791 

30 

31 

.774558 

•905085 

.869473 

.130527 

29 

32 

.774729 

.904992 

.869737 

.130233 

23 

33 

.774899 

.904898 

.870001 

.129999 

27 

31 

.775070 

.904804 

.870265 

.129735 

26 

35 

36 

775240 

.775410 

.904711 

.904617 

.S70529 

.870793 

.129471 

.129207 

25 

24 

37 

.775530 

.904523 

.871057 

.128943 

23 

33 

.775?50 

.904429 

.871321 

.128679 

22 

39 

.775920 

.904335 

.871585 

.128415 

21 

40 

9.776090 

2. S3 
2.83 
2.82 
2.82 
2.82 
2.82 
2.82 
2.82 
2.81 
2.81 

9.904241 

1.57 

1.57 

1.57 

1.57 

1 57 
1.57 
1.57 

1.57 

1.58 
1.53 

9.871849 

4.40 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

0.128151 

20 

41 

42 

43 

.77(3259 

.776429 

.776593 

.904147 

.904053 

.903959 

.872112 

.872376 

.872640 

.127888 

.127624 

.127360 

19 

18 

17 

41 

.776788 

.903864 

.872903 

.127097 

16 

45 

.776937 

,903770 

.873167 

.126833 

15 

46 

.777106 

.903676 

.873430 

.126570 

14 

47 

.777275 

.903581 

.873694 

.126306 

13 

43 

.777444 

.903487 

.873957 

.126043 

12 

49 

.777613 

.903392 

.874220 

.125780 

11 

50 

9.777781 

2.81 
2.81 
2.81 
2.81 
2.80 
2.80 
2.80 
. 2.80 
2.80 
2.79 

9.903298 

1.58 

1.58 

1.58 

1.53 

1.58 

1.58 

1.58 
1.53 

1.59 
1.59 

9.874484 

4.39 

4.39 

4.39 

4.39 

4.38 

4.38 

4.38 

4.33 

4.38 

4.38 

0.125516 

10 

51 

.777950 

.903203 

.874747 

.125253 

9 

52 

.778119 

.903108 

.875010 

.124990 

8 

53 

51 

.778237 

.778455 

.903014 

.902919 

.875273 

.875537 

.124727 

.124463 

7 

6 

55 

.778621 

.902324 

.875S00 

.124200 

5 

56 

.778792 

.902729 

.876063 

.123937 

4 

57 

.778960 

.902634 

.876326 

.123674 

3 

53 

.779128 

.902539 

.876589 

.123411 

2 

59 

.779295 

.902441 

.876352 

.123148 

1 

60 

.779463 

.902349 

.877114 

.122886 

0 

M. 

Cosine. 

1 D. 1". 

Sine. 

D. 1". 

Cctang. 

D. 1". 

Tang. 

M. 

-=j 


133° , 53. 



































COSINES, TANGENTS, AND COTANGENTS 


211 

1433 


373 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.779463 

.779631 

.779798 

.779966 

.780133 

.780300 

.780467 

.780634 

.780801 

.780968 

9.781134 

.781301 

.781468 

.781634 

.781800 

.781966 

.782132 

.782298 

.782464 

.782630 

9.782796 

.782961 

.783127 

.783292 

.783458 

.783623 

.783788 

.783953 

.784118 

.784282 

9.784447 

.784612 

.784776 

.784941 

.785105 

.785269 

.785433 

.785597 

.785761 

.785925 

9.786089 

.786252 

.786416 

.786579 

.786742 

.786906 

.787069 

.787232 

.787395 

.787557 

9.787720 

.787883 

.788045 

.788208 

.788370 

.788532 

.788694 

.788856 

.789018 

.789180 

.789342 

2.79 

2.79 

2.79 

2.79 

2.79 

2.78 

2.78 

2.78 

2.73 
2.78 

2.78 

2.77 

2.77 

2.77 

2.77 

2.77 

2.77 

2.76 

2.76 

2.76 

2.76 

2.76 

2.76 

2.75 

2.75 

2.75 

2.75 

2.75 

2.75 

2.74 

2.74 

2.74 

2.74 

2.74 

2.74 

2.73 

2.73 

2.73 

2.73 

-2.73 

2.73 

2.73 

2.72 

2.72 

2.72 

2.72 

2.72 

2.72 

2.71 

2.71 

2.71 

2.71 

2.71 

2.71 

2.70 

2.70 

2.70 

2.70 

2.70 

2.70 

9.902349 

.902253 

.902158 

.902063 

.901967 

.901872 

.901776 

.901681 

.9015S5 

.901490 

9.901394 

.901293 

.901202 

.901106 

.901010 

.900914 

.900818 

.900722 

.900626 

.900529 

9.900433 

.900337 

.900240 

.900144 

.900047 

.899951 

.899854 

.899757 

.899660 

.899564 

9.899467 

.899370 

.899273 

.899176 

.899078 

.898981 

.898884 

.898787 

.898689 

.898592 

9.898494 

.898397 

.898299 

.898202 

.898104 

.898006 

.897908 

.897810 

.897712 

.897614 

9.897516 

.897418 

.897320 

.897222 

.897123 

.897025 

.896926 

.896828 

.896729 

.896631 

.896532 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 

9.877114 
.877377 
.877640 
.877903 
.S7S165 
.878423 
: .878691 

.878953 
.879216 
.879478 

9.S79741 

.880003 

.880265 

.880528 

.880790 

.881052 

881314 

.881577 

.S81S39 

.882101 

9.8S23G3 

.882625 

.882887 

.883148 

.883410 

.883672 

.833934 

,834196 

.884457 

.884719 

9.8S4980 

.885242 

.885504 

.885765 

.886026 

.8S62SS 

.886549 

.886811 

.8S7072 

.887333 

9.887594 

.887855 

.888115 

.888378 

.888639 

.838900 

.889161 

.889421 

.889682 

.889943 

9.890204 

.890465 

.890725 

.890986 

891247 

.891507 

.891768 

.892028 

.892289 

.892549 

.892810 

4.38 

4.38 

4.33 

4.38 

4.33 

4.38 

4.33 

4.38 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 
4.36 
4.36 
4.36 
4.36 

4.38 
4.36 
4.36 
4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 
4.35 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
f 4.34 

0.122SS6 

.122623 

.122360 

.122097 

.121835 

.121572 

.121309 

.121047 

.1207S4 

.120522 

0.120259 

.119997 

.119735 

.119472 

.119210 

.118948 

.118686 

.118423 

.118161 

.117899 

0.117637 

.117375 

.117113 

.116852 

.116590 

.116328 

.116066 

.115804 

.1155-13 

.115281 

0.115020 

.114758 

.11-1496 

.114235 

.113974 

.113712 

.113451 

.113189 

.112928 

.112667 

0.112406 
.112145 
.111884 
.111622 
.111361 
.111100 
.110839 
. 110579 
.110318 
.110057 

0.109796 
.109535 
.109275 
.109014 
.10S753 
.10S493 
.108232 
.107972 
.107711 
.107451 
.107190 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

45 
47 

46 
45 
44 
43 
42 
41 

40 

39 

38 

37 

38 
35 
34 
33 
32 
31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 1 

D. 1". 

Tang. 

M. 


52 <- 


1270 




























































212 

38° 


1410 


TABLE XIII. LOGARITHMIC SINES, 


M. 

Sine. 

I). 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

9.789342 

2.69 

2.69 

2.69 

2.69 

2.69 

2.69 

2.68 

2.68 

2.68 

2.68 

2.68 

2.68 

2.67 

2.67 

2.67 

2.67 

2.67 

2.67 

2.67 

2.66 

2.66 

2 .66. 

2.66 

2.66 

2.66 

2.65 

2.65 

2.65 

2.65 

2.65 

2.65 

2.64 

2.64 

2.64 

2.64 

2.64 

2.64 

2.64 

2.63 

2.63 

2.63 

2.63 

2.63 

2.63 

2.62 

2.62 

2.62 

2.62 

2.62 

2.61 

2.61 

2.61 

2.61 

2.61 

2.61 

2.61 

2.61 

2.60 

2.60 

2.60 

9.896532 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 

1.69 

1.69 

1.69 

1.69 

1.69 

1.69 

1.69 

1.69 

1.69 

1.69 

1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 

9.892810 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

0.107190 

60 

1 

.789504 

.896433 

.893070 

.106930 

59 

2 

.789665 

.896335 

.893331 

.106669 

58 

3 

.789827 

.896236 

.893591 

.106409 

57 

4 

.789988 

.896137 

.893851 

.106149 

56 

5 

.790149 

.896038 

.894111 

.105889 

55 

6 

.790310 

.895939 

.894372 

.105628 

54 

7 

.790471 

.895840 

.894632 

.105368 

53 

8 

.790632 

.895741 

.894892 

.105108 

52 

9 

10 

.790793 

9.790954 

.895641 

9.895542 

.895152 

9.895412 

.104848 

0.104588 

51 

50 

11 

.791115 

.895443 

.895672 

.104328 

49 

12 

.791275 

.895343 

.895932 

.104068 

48 

13 

.791436 

.895244 

.896192 

.103808 

47 

14 

.791596 

.895145 

.896452 

.103548 

46 

15 

.791757 

.895045 

.896712 

.103288 

45 

16 

.791917 

.894945 

.896971 

.103029 

44 

17 

.792077 

.894846 

.897231 

.102769 

43 

18 

.792237 

.894746 

.897491 

.102509 

42 

19 

20 

.792397 

9.792557 

.894646 

9.894546 

.897751 

9.898010 

.102249 

0.101990 

41 

40 

21 

.792716 

.894446 

.898270 

.101730 

39 

22 

.792376 

.894346 

.898530 

.101470 

38 

23 

.793035 

.894246 

.898789 

.101211 

37 

24 

.793195 

.894146 

.899049 

.100951 

36 

25 

.793354 

.894046 

.899308 

.100692 

35 

26 

.793514 

.893946 

.899568 

.100432 

34 

27 

.793673 

.893846 

.899827 

.100173 

33 

28 

.793832 

.893745 

.900087 

.099913 

32 

29 

30 

.793991 

9.794150 

.893645 

9.893544 

.900346 

9.900605 

.099654 

0.099395 

31 

30 

31 

.794308 

.893444 

.900S64 

.099136 

29 

32 

.794467 

.893343 

.901124 

.098876 

28 

33 

.794626 

.893243 

.901383 

.098617 

27 

34 

.794784 

.893142 

.901642 

.098358 

26 

35 

.794942 

.893041 

.901901 

.098099 

25 

36 

.795101 

.892940 

.902160 

.097840 

24 

37 

.795259 

.892839 

.902420 

.097580 

23 

38 

.795417 

.892739 

.902679 

.097321 

22 

39 

40 

.795575 

9.795733 

.892638 

9.892536 

.902938 

9.903197 

.097062 

0.096803 

21 

20 

41 

.795891 

t .892435 

.903456 

.096544 

19 

42 

.796049 

.892334 

.903714 

.096286 

lS 

43 

.796206 

.892233 

.903973 

.096027 

17 

44 

.796364 

.892132 

.904232 

.095768 

16 

45 

.796521 

.892030 

.904491 

.095509 

15 

46 

.796679 

.891929 

.904750 

.095250 

14 

47 

.796836 

.891827 

.905008 

.094992 

13 

48 

.796993 

.891726 

.905267 

.094733 

12 

49 

50 

.797150 

9.797307 

.891624 

9.891523 

.905526 

9.905785 

.094474 

0.094215 

11 

10 

51 

.797464 

.891421 

.906043 

.093957 

9 

52 

.797621 

.891319 

.906302 

.093698 

8 

53 

.797777 

.891217 

.906560 

.093440 

7 

54 

.797934 

.891115 

.906S19 

.093181 

6 

55 

.79S091 

.891013 

.907077 

.092923 

5 

56 

.798247 

.890911 

.907336 

.092664 

4 

57 

.798403 

.890809 

.907594 

.092406 

3 

58 

.798560 

.890707 

.907853 

.092147 

2 

59 

.798716 

.890605 

.908111 

.091889 

1 

60 

.798872 

.890503 

.908369 

.091631 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


128 ° 


51 









































COSINES, TANGENTS, AND COTANGENTS 


QIC. 

•V It* 

39° 140° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 
27 
23 

29 

30 

31 

32 

33 

34 

35 
38 

37 

38 
39* 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.793S72 

.799023 

.799134 

.799339 

.799495 

.799651 

.799306 

.799962 

.800117 

.800272 

9.800427 

.800582 

.800737 

.800392 

.801047 

.801201 

.801356 

.801511 

.801665 

.801819 

9.801973 

.802123 

.802282 

.802436 

.802589 

.802743 

.802397 

.803050 

.803204 

.803357 

9.803511 

.803664 

.803317 

.803970 

.804123 

.804276 

.804428 

.804581 

.804734 

.804886 

9.805039 

.805191 

.805343 

.805495 

.805647 

.805799 

.805951 

.806103 

.806254 

.806406 

9.806557 
.806709 
.806360 
.807011 
.807163 
.807314 
.807465 
.807615 
.807766 
.807917 
.808067 

2.60 

2.60 

2.60 

2.59 

2.59 

2.59 

2.59 

2.59 

2.59 

2.59 

2.53 
2.58 
2.58 
2.58 
2.58 
2.58 
2.57 
2.57 
2.57 
2.57 

2.57 

2.57 

2.57 

2.56 

2.56 

2.56 

2.56 

2.56 

2.56 

2.55 

2.55 

2.55 

2.55 

2.55 

2.55 

2.55 

2.54 
2.54 
2.54 
2.54 

2.54 

2.54 

2.54 

2.53 

2.53 

2.53 

2.53 

2.53 

2.53 

2.52 

2.52 

2.52 

2.52 

2.52 

2.52 

2.52 

2.51 

2.51 

2.51 

2.51 

9.890503 

.890400 

.890298 

.890195 

.890093 

.889990 

.889838 

.889785 

.889632 

.889579 

9.839477 

.889374' 

.889271 

.889163 

.889064 

.888961 

.888853 

.888755 

.883651 

.888548 

9.888444 

.883341 

.888237 

.888134 

.888030 

.887926 

.887822 

.887718 

.887614 

.887510 

9.887406 

.887302 

.887198 

.887093 

.886989 

.886885 

.886780 

.836676 

.886571 

.886466 

9.886362 

.886257 

.886152 

.886047 

.885942 

.885837 

.885732 

.885627 

.835522 

.885416 

9.885311 

.885205 

.885100 

.884994 

.884889 

.884783 

.8S4677 

.884572 

.884466 

.884360 

.884254 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.72 

1.72 

1.72 

1.72 

1.72 

1.72 

1.72 

1.72 

1.72 

1.72 

1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 

1.73 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.76 

1.76 

1.76 

1.76 

1.76 

1.76 

1.76 

1.76 

1.76 

1.77 
1.77 

9.903369 

.903628 

.903886 

.909144 

.909402 

.909660 

.909918 

.910177 

.910435 

.910693 

9.910951 

.911209 

.911467 

.911725 

.911982 

.912240 

.912498 

.912756 

.913014 

.913271 

9.913529 

.913787 

.914044 

.914302 

.914560 

.914817 

.915075 

.915332 

.915590 

.915847 

9.916104 

.916362 

.916619 

.916877 

.917134 

.917391 

.917648 

.917906 

.918163 

.918420 

9.918677 

.918934 

.919191 

.919448 

.919705 

.919962 

.920219 

.920476 

.920733 

.920990" 

9.921247 

.921503 

.921760 

.922017 

.922274 

.922530 

.922787 

.923044 

.923300 

.923557 

.923814 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.29 

4.29 

.4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.28 

4.28 

4.28 

4.28 

4.23 

4.28 

4.23 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.23 

4.28 

4.28 

4.23 

0.091631 

.091372 

.091114 

.090356 

.090598 

.090340 

.090082 

.089823 

.089565 

.089307 

0.089049 

.0S8791 

.088533 

.088275 

.038018 

.087760 

.037502 

.087244 

.036986 

.086729 

0.086471 

.086213 

.085956 

.085698 

.085440 

.085183 

.084925 

.084668 

.084410 

.084153 

0.0S3S96 

.083638 

.083381 

.083123 

.082S66 

.0S2609 

.082352 

.082094 

.081837 

.081580 

0.081323 

.081066 

.080809 

.080552 

.080295 

.080038 

.079781 

.079524 

.079267 

.079010 

0.078753 

.078497 

.078240 

.077983 

.077726 

.077470 

.077213 

.076956 

.076700 

.076443 

.076186 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


1393 








































214 TABLE XIII. LOGARITHMIC SINES, 


400 139° 


M. 

Sine. 

D 1". 

Cosine. 

D. l'L 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

9.808067 

.808218 

2.51 

2.51 

2.51 

2.50 

2.50 

2.50 

2.50 

2.50 

2.50 

2.50 

9.884254 

.884148 

1.77 

1.77 

1.77 

1.77 

1.77 

1.77 

1.77 

1.77 

1.78 
1.78 

9.923814 

.924070 

4.28 

4.28 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

0.076186 

.075930 

60 

59 

2 

.808368 

.884042 

.924327 

.075673 

58 

3 

.808519 

.883936 

.924583 

.075417 

57 

4 

.808669 

.8S3S29 

.924840 

.075160 

56 

5 

.808819 

.883723 

.925096 

.074904 

55 

6 

7 

.80S969 
.809119 

.833617 

.883510 

.925352 

.925609 

.074648 

.074391 

54 

53 

8 

‘.809269 

.883404 

.925865 

.074135 

52 

9 

.809419 

.883297 

.926122 

.073878 

51 

10 

9.809569 

2.49 

2.49 

2.49 

2.49 

2.49 

2.49 

2.48 

2.48 

2.48 

2.43 

9.883191 

1.78 

1.78 

1.78 

1.78 

1.78 

1.78 

1.78 

1.79 
1.79 
1.79 

9.926378 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

0.073622 

50 

11 

.809718 

.883084 

■ .926634 

.073366 

49 

12 

.809868 

.882977 

.926890 

.073110 

48 

13 

-.810017 

.882871 

.927147 

.072853 

47 

14 

.810167 

.882764 

.927403 

.072597 

46 

15 

.810316 

.882657 

.927659 

.072341 

45 

16 

.810465 

.882550 

.927915 

.072085 

44 

17 

.810614 

.882443 

.928171 

.071829 

43 

18 

.810763 

.882336 

.923427 

.071573 

42 

19 

.810912 

.882229 

.928684 

.071316 

41 

20 

9.811061 

2.43 

2.48 

2.48 

2.47 

2.47 

2.47 

2.47 

2.47 

2.47 

2.47 

9882121 

1.79 

1.79 

1.79 

1.79 

1.79 

1.79 

1.79 

1.80 
1.80 
1.80 

9.928940 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.26 

4.26 

4.26 

4.26 

0.071060 

40 

21 

.811210 

.882014 

.929196 

.070804 

39 

22 

.811358 

.881907 

.929452 

.070548 

38 

23 

.811507 

.881799 

.929708 

.070292 

37 

24 

.811655 

.881692 

.929964 

.070036 

36 

25 

.811804 

.881584 

.930220 

.069780 

35 

26 

.811952 

.881477 

.930475 

.069525 

34 

27 

.812100 

.831369 

.930731 

.069269 

33 

28 

.812248 

.881261 

.930987 

.069013 

32 

29 

.812396 

.881153 

.931243 

.063757 

31 

30 

9.812544 

2.46 

2.46 

2.46 

2.46 

2.46 

2.46 

2.46 

2.45 

2.45 

2.45 

9.881046 

1.80 

1.80 

1.80 

1.80 

1.80 

1.80 

1.81 

1.81 

l.Sl 

1.81 

9.931499 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

0.068501 

30 

31 

312692 

.880933 

.931755 

.068245 

29 

32 

.812340 

.880830 

.932010 

.067990 

28 

33 

.812988 

.880722 

.932266 

.067734 

27 

34 

.813135 

.880613 

.932522 

.067478 

26 

35 

.813283 

.880505 

.932778 

.067222 

25 

36 

.813430 

.880397 

.933033 

.066967 

24 

37 

38 

.813578 

.813725 

.880289 

.880180 

.933289 

.933545 

.066711 

.066455 

23 

22 

39 

.813872 

.8S0072 

.933300 

.066200 ' 

21 

40 

9.814019 

2.45 

2.45 

2.45 

2.45 

2.44 

2.44 

2.44 

2.44 

2.44 

2.44 

9.879963 

1.81 

1.81 

1.81 

1.81 

1.81 

1.81 

1.82 

1.S2 

1.82 

1.82 

9.934056 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

0.065944 

20 

41 

.814166 

.879855 

.934311 

.065689 

19 

42 

.814313 

.879746 

.934567 

.065433 

18 

43 

.814460 

.879637 

.934822 

.065178 

17 

44 

.814607 

.879529 

.935078 

.064922 

16 

45 

.814753 

.879420 

.935333 

.064667 

15 

46 

.814900 

.879311 

.935589 

.064411 

14 

47 

.815046 

.879202 

.935844 

.064156 

13 

48 

49 

.815193 

.815339 

.879093 

.878934 

.936100 

.936355 

.063900 

.063645 

12 

11 

50 

9.815485 

2.44 

2.43 

2.43 

2.43 

2.43 

2.43 

2.43 

2.43 

2.42 

2.42 

9.878875 

1.82 

1.82 

1.82 

1.82 

1.82 

1.83 

1.83 

1.83 

1.83 

1.83 

9.936611 

4 26 
4.26 
4.26 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 

0.0633S9 

10 

51 

.815632 

.878766 

.936866 

.063134 

9 

52 

.815778 

.878656 

.937121 

.062879 

8 

53 

.815924 

.878547 

.937377 

.062623 

7 

54 

55 

.816069 

.816215 

.878438 

.878328 

.937632 

.937887 

.062368 

.062113 

6 

5 

56 

.816361 

.878219 

.938142 

.061858 

4 

57 

.816507 

.878109 

.938398 

.061602 

3 

58 

.816652 

.877999 

.938653 

.061347 

2 

59 

60 

.816798 

.816943 

.877890 

.877780 

.938908 

.939163 

.061092 

.060837 

1 

0 

M. 

Cosine 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 




493 





































410 


COSINES, TANGENTS, AND COTANGENTS 


215 


1380 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1». 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.816943 

.817083 

.817233 

.817379 

.817524 

.817663 

.817813 

.817958 

.818103 

.818247 

9.818392 

.818536 

.818631 

.818825 

.818969 

.819113 

.819257 

.819401 

.819545 

.819689 

9.819832 

.819976 

.820120 

.820263 

.820406 

.820550 

.820693 

.820336 

.820979 

.821122 

9.821265 

.821407 

.821550 

.821693 

.821835 

.821977 

.822120 

.822262 

.822404 

.822546 

9.8226S8 

.822830 

.822972 

.823114 

.823255 

.823397 

.823539 

.823680 

.823821 

.823963 

9.824104 

.824245 

.824386 

.824527 

.824668 

.824808 

.824949 

.825080 

.825230 

.825371 

.825511 

2.42 

2.42 

2.42 

2.42 

2.42 

2.41 

2.41 

2.41 

2.41 

2.41 

2.41 

2.41 

2.40 

2.40 

2.40 

2.40 

2.40 

2.40 

2.40 

2.39 

2.39 

2.39 

2.39 

2.39 

2.39 

2.39 

2.38 

2.38 

2.38 

2.33 

2.38 

2.38 

2.38 

2.37 

2.37 

2.37 

2.37 

2.37 

2.37 

2.37 

2.37 

2.36 

2.36 

2.36 

2.36 

2.36 

2.36 

2.36 

2.35 

2.35 

2.35 

2.35 

2.35 

2.35 

2.35 

2.34 
2.34 
2.34 
2.34 
2.34 

9.877780 

.877670 

.877560 

.877450 

.877340 

.877230 

.877120 

.877010 

.876899 

.876789 

9.876678 

.876568 

.876457 

.876347 

.876236 

.876125 

.876014 

.875904 

.875793 

.875682 

9.875571 

.875459 

.875348 

.875237 

.875126 

.875014 

.874903 

.874791 

.874680 

.874563 

9.874456 

.874344 

.874232 

.874121 

.874009 

.873896 

.873784 

.873672 

.873560 

.873448 

9.873335 

.873223 

.873110 

.872998 

.872885 

.872772 

.872659 

.872547 

.S72434 

.872321 

9.872208 
.872095 
* .871981 
.871868 
.871755 
.871641 
.871528 
.871414 
.871301 
.871187 
.871073 

1.83 

1.83 

1.83 

1.83 

1.84 
1.84 
1.84 
1.84 
1.84 
1.84 

1.84 

1.84 

1.84 

1.84 

1.85 
1.85 
1.85 
1.85 
1.85 
1.85 

1.85 

1.85 

1.85 

1.86 
1.86 
1.86 
1.86 
1.86 
1.86 
1.86 

1.86 

1.86 

1.87 

1.87 

1.87 

1.87 

1.87 

1.87 

1.87 

1.87 

1.87 

1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1 .8S 
1.88 
1.88 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.90 

9.939163 

.939418 

.939673 

.939928 

.940183 

.940439 

.940694 

.940949 

.941204 

.941459 

9.941713 

.941968 

.942223 

.942478 

.942733 

.942988 

.943243 

.943498 

.943752 

.944007 

9.944262 

.944517 

.944771 

.945026 

.945281 

.945535 

.945790 

.946045 

.946299 

.946554 

9.946808 

.947063 

.947318 

.947572 

.947827 

.948081 

.948335 

.948590 

.948844 

.949099 

9.949353 

.949608 

.949S62 

.950116 

.950371 

.950625 

.950879 

.951133 

.951388 

.951642 

9.951896 

.952150 

.952405 

.952659 

.952913 

.953167 

.953421 

.953675 

.953929 

.954183 

.954437 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.23 

4.23 

4.23 

0.060S37 
.060582 
.060327 
.060072 
.059817 
.059561 
.059306 
.059051 
.058796 
.058541 

0.058287 

.058032 

.057777 

.057522 

.057267 

.057012 

.056757 

.056502 

.056248 

.055993 

0.055738 

.055483 

.055229 

.054974 

.054719 

.054465 

.054210 

.053955 

.053701 

.053446 

0.053192 

.052937 

.052682 

.052428 

.052173 

.051919 

.051665 

.051410 

.051156 

.050901 

0.050647 

.050392 

.050138 

.049884 

.049629 

.049375 

.049121 

.048867 

.048612 

.048358 

0.048104 

.047850 

.047595 

.047341 

.047087 

.046833 

.046579 

.046325 

.046071 

.045817 

.045563 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine.- 

D. 1". 

Cotang. 1 

D. 1". 

Tang. 

M. 


480 


1310 








































216 

43 ° 


TABLE XIII. LOGARITHMIC SINES, 

13>j-o 


M. 

Sine. 

D. 1»‘. 

Cosine. 

D. 1". 

Tang. 

D. 1". 

I 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 ’ 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.825511 
.825651 
.825791 
.825931 
.826071 
.826211 
.826351 
.826491 
.826631 
.826770 

9.826910 

.827049 

.827189 

.827328 

.827467 

.827606 

.827745 

.827884 

.823023 

.823162 

9.828301 

.823439 

.828578 

.828716 

.823855 

.828993 

.829131 

.829269 

.829407 

.829545 

9.829633 

.829821 

.829959 

.830097 

.830234 

830372 

.830509 

.830646 

.830784 

.830921 

9.831058 

.831195 

.831332 

.831469 

.831606 

.831742 

.831879 

.832015 

.832152 

.832288 

9.832425 

.832561 

.832697 

.832333 

.832969 

.833105 

.833241 

.833377 

.833512 

.833648 

.833783 

2.34 

2.34 

2.33 

2.33 

2.33 

2.33 

2.33 

2.33 

2.33 

2.33 

2.32 

2.32 

2.32 

2.32 

2.32 

2.32 

2.32 

2.31 

2.31 

2.31 

2.31 

2.31 

2.31 

2.31 

2.31 

2.30 

2.30 

2.30 

2.30 

2.30 

2.30 

2.30 

2.29 

2.29 

2.29 

2.29 

2.29 

2.29 

2.29 

2.29 

2.23 

2.28 

2.23 

2.28 

2.28 

2.28 

2.23 

2.27 

2.27 

2.27 

2.27 

2.27 

2.27 

2.27 

2.27 

2.26 

2.26 

2.26 

2.26 

2.26 

9.871073 

.870960 

.870346 

.870732 

.870618 

.870504 

.870390 

.870276 

.870161 

.870047 

9.869933 

.869818 

.869704 

.869589 

.869474 

.869360 

.869245 

.869130 

.869015 

.863900 

9.863785 

.868670 

.868555 

.863440 

.863324 

.863209 

.863093 

.867978 

.867862 

.867747 

9.867631 

.867515 

.867399 

.867283 

.867167 

.867051 

.866935 

.866819 

.866703 

.866586 

9.866470 

.866353 

.866237 

.866120 

.866004 

.865387 

.865770 

.865653 

.865536 

.865419 

9.865302 

.865185 

.865063 

.864950 

.864333 

.864716 

.864593 

.864481 

.864363 

.864245 

.864127 

1.90 

1.90 

1.90 

1.90 

1.90 

1.90 

1.90 

1.90 

1.91 
1.91 

1.91 

1.91 

1.91 

1.91 

1.91 

1.91 

1.91 

1.92 
1.92 
1.92 

1.92 

1.92 

1.92 

1.92 

1.92 

1.92 

1.93 
1.93 
1.93 
1.93 

1.93 

1.93 

1.93 

1.93 

1.93 

1.94 
1.94 
1.94 
1.94 
1.94 

1.94 

1.94 

1.94 

1 94 

1.95 
1.95 
1.95 
1.95 
1.95 
1.95 

1.95 

1.95 

1.95 

1.96 
1.96 
1.96 
1.96 
1.96 
1.96 
1.96 

9.954437 

.954691 

.954946 

.955200 

.955454 

.955708 

.955961 

.956215 

.956169 

.956723 

9.956977 

.957231 

.957485 

.957739 

.957993 

.958247 

.953500 

.953754 

.959003 

.959262 

9.959516 

.959769 

.960023 

.960277 

.960530 

.960784 

.961038 

.961292 

.961545 

.961799 

9.962052 

.962306 

.962560 

.962813 

.963067 

.963320 

.963574 

.963828 

.964081 

.964335 

9.964588 
.964342 
.965095 
.965349 
.965602 
.965855 
.966109 
.966362 
.9666!6 
.966869 

9.967123 

.967376 

.967629 

.967883 

.968136 

.968389 

.963643 

.963896 

.969149 

.969403 

.969656 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.22' 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

0.045563 

.045309 

.045054 

.044800 

.044546 

.044292 

.044039 

.043785 

.043531 

.043277 

0.043023 

.042769 

.042515 

.042261 

.042007 

.041753 

.041500 

.041246 

.040992 

.040738 

0.040484 

.0402.31 

.039977 

.039723 

.039470 

.039216 

.038962 

.038708 

.038455 

.038201 

0.037948 

.037694 

.037440 

.037187 

.036933 

.036680 

.036426 

.036172 

.035919 

.035665 

0.035412 

.035158 

.034905 

.034651 

.034393 

.034145 

.033391 

.033633 

.033384 

.033131 

0.032877 

.032624 

.032371 

.032117 

.031864 

.031611 

.031357 

.031104 

.030351 

.030597 

.030344 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

23 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1’. 

Tang. 

M. 


133 ° 































COSINES, TANGENTS, AND COTANGENTS 


217 

136° 


43 ° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.833783 

.833919 

.834054 

.834189 

.834325 

.834460 

.831595 

.834730 

.834865 

.834999 

9.835134 

.835269 

.835403 

.835538 

.835672 

.835807 

.835941 

.836075 

.836209 

.836343 

9.836477 

.836611 

.836745 

.836878 

.837012 

.837146 

.837279 

.837412 

.837546 

.837679 

9.837812 

.837915 

.838078 

.833211 

.838344 

.838477 

.838610 

.838742 

.838875 

.839007 

9.839140 

.839272 

.839404 

.839536 

.839668 

.839800 

.839932 

.840064 

.840196 

.840323 

9.840459 

.840591 

.840722 

.840854 

.840985 

.841116 

.841247 

.841378 

.841509 

.841640 

.841771 

2.26 

2.26 

2.25 

2.25 

2.25 

2.25 

2.25 

2.25 

2.25 

2.25 

2.24 

2.24 

2.24 

2.24 

2.24 

2.24 

2.24 

2.23 

2.23 

2.23 

2.23 

2.23 

2.23 

2.23 

2.23 

2.22 

2,22 

2.22 

2.22 

2.22 

2.22 

2.22 

2.22 

2.21 

2.21 

2.21 

2.21 

2.21 

2.21 

2.21 

2.21 

2.20 

2.20 

2.20 

2.20 

2.20 

2.20 

2.20 

2.19 

2.19 

2.19 

2.19 

2.19 

2.19 

2.19 

2.19 

2.18 

2.18 

2.18 

2.18 

9.864127 

.864010 

.863892 

.863774 

.863656 

.883533 

.863419 

.863301 

.863183 

.863064 

9.862946 

.862827 

.862709 

.862590 

.862471 

.862353 

.862234 

.862115 

.861996 

.861877 

9.861758 

.861638 

.861519 

.861400 

.861280 

.861161 

.861041 

.860922 

.860802 

.860682 

9.860562 

.860442 

.860322 

.860202 

.860082 

.859962 

.859842 

.859721 

.859601 

.859480 

9.859360 

.859239 

.859119 

.858998 

.858877 

.858756 

.858635 

.858514 

.858393 

.858272 

9.858151 
.858029 
.857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.857178 
f .857056 
.856934 

1.98 

1.97 

1.97 

1.97 

1.97 

1.97 

1.97 

1.97 

1.97 

1.97 

1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 

1.98 

1.99 

1.99 

1.99 

1.99 

1.99 

1.99 

1.99 

1.99 

2.00 

2.00 

2,00 

2.00 

2.00 

2.00 

2.00 

2.00 

2.00 

2.01 

2.01 

2.01 

2.01 

2.01 

2.01 

2.01 

2.01 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.03 

2.03 

2.03 

2.03 

2.03 

2.03 

2.03 

9.969656 

.969909 

.970162 

.970416 

.970669 

.970922 

.971175 

.971429 

.971682 

.971935 

9.972188 

.972441 

.972695 

.972948 

.973201 

.973454 

.973707 

.973960 

.974213 

.974466 

9.974720 

.974973 

.975226 

.975479 

.975732 

.9759S5 

.97623S 

.976491 

.976744 

.976997 

9.977250 

.977503 

.977756 

.978009 

.978262 

.978515 

.97S768 

.979021 

.979274 

.979527 

9.979780 

.980033 

.980286 

.980538 

.980791 

.981044 

.981297 

.981550 

.981803 

.982056 

9.982309 

.982562 

.9S2814 

.983067 

.983320 

.983573 

.983826 

.984079 

.984332 

.984584 

.984837 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22' 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21* 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

0.030344 

.030091 

.029838 

.029584 

.029331 

.029078 

.028825 

.028571 

.028318 

.028065 

0.027812 

.027559 

.027305 

.027052 

.026799 

.026546 

.026293 

.026040 

.025787 

.025534 

0.025280 

.025027 

.024774 

.024521 

.024268 

.024015 

.023762 

.023509 

.023256 

.023003 

0.022750 

.022497 

.022244 

.021991 

.021738 

.021485 

.021232 

.020979 

.020726 

.020473 

0.020220 

.019967 

.019714 

.019462 

.019209 

.018956 

.018703 

.018450 

.018197 

.017944 

0.017691 

.017438 

.017186 

.016933 

.016680 

.016427 

.016174 

.015921 

.015668 

.015416 

.015163 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Taug. 

M. 


133 ° 





































218 

440 


TABLE XIII. LOGARITHMIC BINES, &C. 

135 ° 


M. 

Sine. 

D. 1".' 

Cosine. 

D. 1". 

Tang. 

B. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 
27 
23 

29 

30 

31 

32 

33 

34 

35 

38 
37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
53 

59 

60 

9.841771 

.841902 

.842033 

.842163 

.842294 

.842424 

.842555 

.842685 

.842815 

.842946 

9.843076 

.843206 

.843336 

.843466 

.843595 

.843725 

.843855 

.843934 

.844114 

.844243 

9.844372 

.844502 

.844631 

.844760 

.844889 

.845018 

.845147 

.845276 

.845405 

.845533 

9.845662 

.845790 

.845919 

.846047 

.846175 

.846304 

.816432 

.846560 

.846633 

.846316 

9.846944 

.847071 

.847199 

.847327 

.847454 

.847582 

.847709 

.847836 

.847964 

.843091 

9.843218 

.848345 

.843472 

.843599 

.848726 

.848852 

.843979 

.849106 

.849232 

.849359 

.849485 

2.18 

2.18 

2 18 
2.18 
2.17 
2.17 
2.17 
2.17 
2.17 
2.17 

2.17 
2.17 
2.16 
■ 2.16 
2.16 
2.16 
2.16 
2.16 
2.16 
2.16 

2.15 

2.15 

2.15 

2.15 

2.15 

2.15 

2.15 

2.15 

2.14 

2.14 

2.14 

2.14 

2.14 

2.14 

2.14 

2.14 

2.13 

2.13 

2.13 

2.13 

2.13 

2.13 

2.13 

2.13 

2.12 

2.12 

2.12 

2.12 

2.12 

2.12 

2.12 

2.12 

2.11 

2.11 

2.11 

2.11 

2.11 

2.11 

2.11 

2.11 

9.856934 

.856812 

.856690 

.856563 

.856446 

.856323 

.856201 

.856078 

.855956 

.855833 

9.855711 

.855588 

.855465 

.855342 

.855219 

.855096 

.854973 

.854850 

.854727 

.854603 

9.854430 

.854356 

.854233 

.854109 

.8539S6 

.853362 

.853738 

.853614 

.853490 

.853366 

9.853242 

.853118 

.852994 

.852869 

.852745 

.852620 

.852496 

.852371 

.852247 

.852122 

9.851997 

.851872 

.851747 

.851622 

.851497 

.851372 

.851246 

.851121 

.850996 

.850370 

9.850745 

.850619 

.850493 

.850363 

.850242 

.850116 

.849990 

.849364 

.849738 

.849611 

.849435 

2.03 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.07 

2.07 

2.07 

2.07 

2.07 

2.07 

2.07 

2.07 

2.03 

2.03 

2.03 

2.08 

2.03 

2.03 

2.03 

2.08 

2.09 

2.09 

2.09 

2.09 

2.09 

2.09 

2.09 

2.09 

2.10 

2.10 

2.10 

2.10 

2.10 

2.10 

2.10 

2.10 

2.11 

9.934837 

.985090 

.935343 

.935596 

.935348 

.936101 

.986354 

.986607 

.986860 

.987112 

9.937365 

.937618 

.987371 

.988123 

.9S8376 

.938629 

.988882 

.939134 

.939387 

.9S9640 

9.989893 

.990145 

.990393 

.990651 

.990903 

.991156 

.991409 

.991662 

.991914 

.992167 

9.992420 

.99267,£ 

.992925 

.993178 

.993431 

.993633 

.993936 

.994189 

.994441 

.994694 

9.994947 
.995199 
..995452 
.995705 
.995957 
.996210 
.996463 
.996715 
.99696S 
.997221 

9.997473 

.997726 

.997979 

.998231 

.998484 

.993737 

.993939 

.999242- 

.999495 

.999747 

o.oooooo 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.2) 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

0.015163 

.014910 

.014657 

.014404 

.014152 

.013899 

.013646 

.013393 

.013140 

.012388 

0.012635 

.012332 

.012129 

.011877 

.011624 

.011371 

.011118 

.010866 

.010613 

.010360 

0.010107 

.009355 

.009602 

.009349 

.009097 

.003844 

.003591 

.003333 

.008036 

.007833 

0.007580 

.007323 

.007075 

.006322 

.006569 

.006317 

.006064 

.005811 

.005559 

.005306 

0.005053 

.004801 

.004548 

.004295 

.004043 

.003790 

.003537 

.003285 

.003032 

.002779 

0.002527 

.002274 

.002021 

.001769 

.001516 

.001263 

.001011 

.000758 

.000505 

.000253 

.000000 

60 

59 

53 

57 

56 

55 

54 

53 

52 

51 

50 

49 

43 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

l 

0 

M. | 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


4*50 


134-o 










































TABLE XIV. 


NATURAL SINES AND 


COSINES, 


220 TABLE XIV. NATURAL SINES AND COSINES 



03 

1° 

i 30 

| 30 

40 

—V 

M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

j Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

M. 

0 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

60 

1 

.00029 

One. 

.01774 

.99934 

,03519 

.99938 

.05263 

.99861 

.07005 

.99754 

59 

2 

.00058 

One. 

.01803 

.99934 

.03548 

.99937 

.05292 

.99S60 

.07034 

.99752 

58 

3 

.00037 

One. 

.01832 

.99933 

.03577 

.99936 

05321 

.99858 

.07063 

.99750 

57 

4 

.00116 

One. 

.01862 

.99933 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.99932 

.03635 

.99934 

.05379 

.99855 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99932 

.03664 

.99933 

.05403 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

53 . 

8 

.00233 

One. 

.01973 

.99930 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

.00262 

One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738 

51 

10 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99344 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.03363 

.99925 

.05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03397 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

.99725 

45 

16 

.00465 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99S33 

.07440 

.99723 

44 

17 

.00495 

.99999 

.02240 

.99975 

.03984 

.99921 

.05727 

.99336 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99993 

.02293 

.99974 

.04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00532 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99329 

.07585 

.99712 

39 

22 

.00640 

.99993 

.02335 

.99972 

.04129 

.99915 

.05873 

.99827 

.07614 

.99710 

33 

23 

.00669 

.99993 

.02414 

.99971 

.04159 

.99913 

.05902 

.99326 

.07643 

.99708 

37 

24 

.00698 

99993 

.02443 

.99970 

.04188 

.99912 

.05931 

.99324 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99963 

.04275 

.99909 

.06018 

.99319 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817 

.07788 

.99696 

32 

29 

.00844 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815 

.07817 

.99694 

31 

30 

.00373 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99312 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99310 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99635 

17 

34 

.00939 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99304 

.07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99961 

.04536 

.99397 

.06279 

.99803 

.08020 

.99678 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

.99896 

.06308 

.99801 

.08049 

.99676 

23 

38 

.01105 

.99994 

.02350 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673 

22 

39 

.01134 

.99994 

.02379 

.99959 

.04623 

.99393 

.06366 

.99797 

.08107 

.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04632 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99S88 

.06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

.99657 

15 

46 

.01338 

.99991 

.03033 

.99952 

.04827 

.99S8.3 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.03339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99381 

.06627 

.99780 

.03363 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

.04914 

.99379 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.99939 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03223 

.99948 

.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99939 

.03257 

.99947 

.05001 

.99375 

.06743 

.99772 

.03484 

.99639 

8 

53 

.01542 

.99938 

03236 

.99946 

.05030 

.99373 

.06773 

.99770 

.08513 

.99637 

7 

54 

.01571 

.99938 

.03316 

.99945 

.05059 

.99872 

.06802 

.9976S 

.08542 

.99635 

6 

55 

.01600 

.99937 

03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.01629 

.99937 

.03374 

.99943 

.05117 

.99869 

.06360 

.99764 

.08690 

.99630 

4 

57 

.01658 

.99936 

.03403 

.99942 

.05146 

.99367 

.06389 

.99762 

.08629 

.99627 

3 

53 

.01687 

.99936 

.03432 

.99941 

.05175 

.99S66 

.06918 

.99760 

.08658 

.99625 

2 

59 

.01716 

.99935 

.03461 

.99940 

.05205 

.99364 

.06947 

.99758 

.08687 

.99622 

1 

60 

.01745 

.99985 

.03490 

.99939 

.05234 

.99363 

.06976 

.99756 

.08716 

.99619 

0 

M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

M. 


893 

883 

870 

860 

850 
















































TABLE XIV. NATURAL SINES AND COSINES. 221 



50 

60 

70 

jj 

90 

n 

M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

| Sine. 

Cosin. 

Sine. 

Cosin. 

M. 

0 

.03716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

60 

1 

.03745 

.99617 

.10482 

.99419 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764 

59 

2 

.03774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.98760 

58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4 

.08331 

.99609 

.10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15758 

.98751 

56 

5 

.03S60 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.15787 

.93746 

55 

6 

.08339 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15816 

.98741 

54 

7 

.03918 

.99602 

. 10655 

.99431 

.12389 

.99230 

.14119 

.98998 

.15845 

.98737 

53 

S 

.03947 

.99599 

.10684 

.99423 

.12418 

.99226 

.14148 

.98994 

.15873 

.98732 

52 

9 

.03976 

.99596 

.10713 

.99424 

.12447 

.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

.15959 

.98718 

49 

12 

.09063 

.99588 

.10300 

.99415 

.12533 

.99211 

.14263 

.98978 

.15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412 

.12562 

.99203 

.14292 

.98973 

.16017 

.98709 

47 

14 

.09121 

.99583 

.10358 

.99409 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

46 

15 

.09150 

.99580 

.10387 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700 

45 

16 

.09179 

.99578 

.10916 

.99402 

.12649 

.99197 

.14378 

.98961 

.16103 

.98695 

44 

17 

.09203 

.99575 

.10945 

.99399 

.12678 

.99193 

.14407 

.98957 

.16132 

.98690 

43 

18 

.09237 

.99572 

.10973 

.99396 

.12706 

.99189 

.14436 

.98953 

.16160 

.98636 

42 

19 

.09266 

.99570 

.11002 

.99393 

.12735 

.99186 

.14464 

.93948 

.16189 

.98631 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493 

.93944 

.16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.99562 

.11039 

.99333 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667 

33 

23 

.09332 

•99559 

.11118 

.99330 

.12851 

.99171 

.14580 

.98931 

.16304 

.98662 

37 

24 

.09411 

.99556 

.11147 

.99377 

.12880 

.99167 

.14603 

.98927 

.16333 

.98657 

36 

25 

.09440 

.99553 

.11176 

.99374 

.12903 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.9S919 

.16390 

.98648 

34 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.93914 

.16419 

.98643 

33 

23 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.14723 

.98910 

.16447 

.98638 

32 

29 

.09556 

.99542 

.11291 

.99360 

.13024 

.99143 

.14752 

.98906 

.16476 

.98633 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.93902 

.16505 

.98629 

30 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.14838 

.98893 

.16562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99523 

.11436 

.99344 

.13163 

.99129 

.14896 

.98884 

.16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.93604 

25 

36 

.09753 

.99523 

.11494 

.99337 

.13226 

.99122 

.14954 

.98876 

.16677 

.98600 

24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595 

23 

33 

.09316 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590 

22 

39 

.09345 

.99514 

.11530 

.99327 

.13312 

.99110 

.15040 

.93863 

.16763 

.98585 

21 

40 

.09374 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102 

.15097 

.93854 

.16320 

.98575 

19 

42 

.09932 

.99506 

.11667 

.99317 

.13399 

.99093 

.15126 

.98849 

.16349 

.98570 

18 

43 

.09961 

.99503 

.11696 

.99314 

.13427 

.99094 

.15155 

.93845 

.16378 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13456 

.99091 

.15184 

.93841 

.16906 

.93561 

16 

45 

.10019 

.99497 

.11754 

.99307 

.13485 

.99037 

.15212 

.93836 

.16935 

.98556 

15 

' 46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.15241 

.98832 

.16964 

.98551 

14 

47 

.10077 

.99491 

.11312 

.99300 

.13543 

.99079 

.15270 

.98827 

.16992 

.93546 

13 

48 

.10106 

.99488 

.11340 

.99297 

.13572 

.99075 

.15299 

.98823 

.17021 

.93541 

12 

49 

.10135 

.99435 

.11869 

.99293 

.13600 

.99071 

.15327 

.98318 

.17050 

.98536 

11 

50 

.10164 

.99482 

.11893 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.93531 

10 

51 

.10192 

.99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11956 

.99233 

.13637 

.99059 

.15414 

.98805 

.17136 

.98521 

8 

53 

.10250 

.99473 

.11935 

.99279 

.13716 

.99055 

.15442 

.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.15471 

.93796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

15500 

.93791 

.17222 

.98506 

5 

56 

.10337 

.99464 

.12071 

.99269 

.13302 

.99043 

.15529 

.93787 

.17250 

.98501 

4 

57 

.10366 

.99461 

.12100 

.99265 

.13331 

.99039 

.15557 

.98782 

.17279 

.98496 

3 

53 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

.15586 

.93778 

.17308 

.98491 

2 

59 

.10424 

.99455 

.12158 

.99253 

.13339 

.99031 

.15615 

.98773 

.17336 

.98486 

1 

60 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

.17365 

.98481 

0 

M. 

Cosin. 

Sine. 

Cosin.. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 1 

M. 


84to 

83^ 

83= 

810 

8<P 














































222 TABLE XI 1 ? NATURAL SINES AND COSINES 



103 

lie 

130 

! 130 

140 

1 

M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

[Sine. 

Cosin. 

Sine. 

Cosin. 

M. 

0 

.17365 

.98431 

.19081 

.93163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

60 

1 

.17393 

.93476 

.19109 

.98157 

.20820 

.97809 

.22523 

.97430 

.24220 

.97023 

59 

2 

.17422 

.93471 

.19133 

.93152 

.20843 

.97803 

.22552 

.97424 

.24249 

.97015 

58 

3 

.17451 

.9S466 

.19167 

.98146 

.20377 

.97797 

.22580 

.97417 

.24277 

.97008 

57 

4 

.17479 

.98461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5 

.17503 

.93455 

.19224 

.98135 

.20933 

.97784 

.22637 

.97404 

.24333 

.98994 

55 

6 

.17537 

.93450 

.19252 

.98129 

.20962 

.97778 

.22665 

.97398 

.24362 

.96987 

54 

7 

.17565 

.98445 

.19231 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96930 

53 

8 

.17594 

.93440 

.19309 

.98118 

.21019 

.97766 

.22722 

.97334 

.24418 

.96973 

52 

9 

.17623 

.98435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10 

.17651 

.93430 

.19366 

.93107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.17630 

.98425 

.19395 

.98101 

.21104 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17703 

.93420 

.19423 

.98096 

.21132 

.97742 

.22335 

.97358 

.24531 

.96945 

48 

13 

.17737 

.93414 

.19452 

.98090 

.21161 

.97735 

.22863 

.97351 

.24559 

.96937 

47 

14 

.17766 

.93409 

.19481 

.98034 

.21189 

.97729 

.22392 

97345 

.24587 

.96930 

46 

15 

.17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45 

16 

.17823 

.98399 

.19533 

.98073 

,2i246 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

17 

.17852 

.93394 

.19566 

.98067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

18 

.17830 

.98389 

.19595 

.93061 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

.93333 

.19623 

.93056 

.21331 

.97693 

.23033 

.97311 

.24728 

.96394 

41 

20 

.17937 

.98378 

.19652 

.93050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

.17966 

.93373 

.19630 

.93044 

.21338 

.97686 

.23090 

.97293 

.24784 

.96880 

39 

22 

.17995 

.93363 

.19709 

.93039 

.21417 

.97680 

.23118 

.97291 

.24813 

.96373 

33 

23 

.18023 

.93362 

.19737 

.98033 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

24 

.18052 

.98357 

.19766 

.93027 

.21474 

.97667 

.23175 

.97278 

.24S69 

.96858 

36 

25 

.18081 

.93352 

.19791 

.93021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.93347 

.19823 

.93016 

.21530 

.97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18133 

.93341 

.19851 

.98010 

.21559 

.97648 

.23260 

.97257 

.24954 

.96837 

33 

23 

.18166 

.93336 

.19330 

.93004 

.21587 

.97642 

.23288 

.97251 

.24932 

.96329 

32 

29 

.18195 

.93331 

.19903 

.97998 

.21616 

.97636 

.23316 

.97244 

.25010 

.96822 

31 

30 

.18224 

.93325 

.19937 

.97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

.97937 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18231 

.98315 

.19994 

.97931 

.21701 

.97617 

.23401 

.97223 

.25094 

.96800 

28 

33 

.13309 

.93310 

.20022 

.97975 

.21729 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18333 

.93304 

.20051 

.97969 

.21758 

.97604 

.23458 

.97210 

.25151 

.96786 

26 

35 

.18367 

.93299 

.20079 

.97963 

.21786 

.97598 

.23486 

.97203 

.25179 

.96778 

25 

36 

.18395 

.93294 

.20103 

.97958 

.21814 

.97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

.18424 

.93288 

.20136 

.97952 

.21843 

.97535 

.23542 

.97189 

.25235 

.96764 

23 

33 

.18452 

.93233 

.20165 

.97946 

.21871 

.97579 

.23571 

.97182 

.25263 

.96756 

22 

39 

.18481 

.98277 

.20193 

.97940 

.21899 

.97573 

.23599 

.97176 

.25291 

.96749 

21 

40 

.18509 

.93272 

.20222 

.97934 

.21928 

.97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

. 18538 

.93267 

.20250 

.97923 

.21956 

.97560 

.23656 

.97162 

.25348 

.96734 

19 

42 

.18567 

.98261 

.20279 

.97922 

.21985 

.97553 

.23684 

.97155 

.25376 

.96727 

18 

43 

.18595 

.93256 

.20307 

.97916 

.22013 

.97547 

.23712 

.97148 

.25404 

.96719 

17 

44 

.18624 

.98250 

.20336 

.97910 

.22041 

.97541 

.23740 

.97141 

.25432 

.96712 

16 

45 

.18652 

.93245 

.20364 

.97905 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705 

15 

46 

.18631 

.93240 

.20393 

.97899 

.22098 

.97528 

.23797 

.97127 

.25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

.97893 

.22126 

.97521 

.23325 

.97120 

.25516 

.96690 

13 

43 

.18733 

.98229 

.20450 

.97887 

.22155 

.97515 

.23S53 

.97113 

.25545 

.96632 

12 

49 

.13767 

.93223 

.20478 

.97881 

.22183 

.97503 

23882 

.97106 

.2557.3 

.96675 

11 

50 

.18795 

.93218 

.20507 

.97875 

.22212 

.97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

.18824 

.98212 

.20535 

.97869 

.22240 

.97496 

.23938 

.97093 

.25629 

.96660 

9 

52 

.18852 

.98207 

.20563 

.97863 

.22268 

.97489 

.23966 

.97036 

.25657 

.96653 

8 

53 

.18831 

.93201 

.20592 

.97857 

.22297 

.97483 

.23995 

.97079 

.25635 

.96645 

7 

54 

.18910 

.93196 

.20620 

.97351 

.22325 

.97476 

.24023 

.97072 

.25713 

.96633 

6 

55 

.18933 

.98190 

.20649 

.97345 

.22353 

.97470 

.24051 

.97065 

.25741 

.96630 

5 

56 

18967 

.98185 

.20677 

.97839 

.22382 

.97463 

.24079 

.97058 

.25769 

.96623 

4 

57 

.18995 

.93179 

.20706 

.97833 

.22410 

.97457 

.24103 

.97051 

.25798 

.96615 

3 

53 

.19024 

.93174 

.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826 

.96608 

2 

59 

.19052 

.93163 

.20763 

.97821 

.22467 

.97444 

.24164 

.97037 

.25854 

.96600 

1 

60 

.19081 

.93163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.25882 

.96593 

0 

M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosir. 

Sine. 

Cosin. 

Sine. 

M. 


79 ° 

78 ? 

77 o 

760 

75° 

















































TABLE XIV. NATURAL SINES AND COSINES 


223 



153 

163 

170 

18 J 

190 


M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. M. 

0 

.25832 

.96593 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

60 

l 

.25910 

.96585 

.27592 

.96118 

.29265 

.95622 

.30929 

.95097 

.32584 

.94542 

59 

2 

.25933 

.96578 

.27620 

.96110 

.29293 

.95613 

.30957 

.95088 

.32612 

.94533 

58 

3 

.25966 

.96570 

.27648 

.96102 

.29321 

.95605 

.30985 

.95079 

.32639 

.94523 

57 

4 

.25994 

.96562 

.27676 

.96094 

.29348 

.95596 

.31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.27704 

.96036 

.29376 

.95588 

.31040 

.95061 

.32694 

.94504 

55 

6 

.26050 

.96547 

.27731 

.96078 

.29404 

.95579 

-.31063 

.95052 

.32722 

.94495 

54 

7 

.26079 

.96540 

.27759 

.96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.26107 

.96532 

.27787 

.96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.96524 

.27815 

.96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

.28163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191' 

.96509 

.27871 

.96037 

.29543 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.95523 

.31233 

.94997 

.32887 

.94438 

48 

13 

.26247 

.96494 

.27927 

.96021 

.29599 

.95519 

.31261 

.94988 

.32914 

.94428 

47 

14 

.26275 

.96436 

.27955 

.96013 

.29626 

.95511 

.31289 

.94979 

.32942 

.94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316 

.94970 

.32969 

.94409 

45 

16 

.26331 

.96471 

.23011 

.95997 

.29682 

.95493 

.31314 

.94961 

.32997 

.94399 

44 

17 

.26359 

.96463 

.23039 

.95989 

.29710 

.95435 

.31372 

.94952 

.33024 

.94390 

43 

18 

.26337 

.96456 

.23067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.26415 

.96448 

.23095 

.95972 

.29765 

.95467 

.31427 

.94933 

.33079 

.94370 

41 

20 

.26443 

.96440 

.23123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

.26471 

.96433 

.23150 

.95956 

.29821 

.95450 

.31482 

.94915 

.33134 

.94351 

39 

22 

.26500 

.96425 

.28178 

.95943 

.29349 

.95441 

.31510 

.94906 

.33161 

.94342 

38 

23 

.26523 

.96417 

.23206 

.95940 

.29876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

24 

.26556 

.96410 

.23234 

.95931 

.29904 

95424 

.31565 

.94888 

.33216 

.94322 

36 

25 

.26584 

.96402 

.23262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

.26612 

.96394 

.23290 

.93915 

.29960 

.95407 

.31620 

.94869 

.33271 

.94303 

34 

27 

.26640 

.96386 

.28318 

.95907 

.29987 

.95398 

.31648 

.94860 

.33298 

.94293 

33 

23 

.26668 

.96379 

.23346 

.95893 

.30015 

.95389 

.31675 

.94851 

.33326 

.94284 

32 

29 

.26696 

.96371 

.23374 

.95890 

.30043 

.95380 

.31703 

.94842 

.33353 

.94274 

31 

30 

.26724 

.96363 

.23402 

.95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

.26752 

.96355 

.23429 

.95874 

.30093 

.95363 

.31758 

.94823 

.334OS 

.94254 

29 

32 

.26780 

.96347 

.23457 

.95865 

.30126 

.95354 

.31786 

.94814 

.33436 

.94245 

28 

33 

.26303 

.96340 

.23485 

.95857 

.30154 

.95345 

.31813 

.94805 

.33463 

.94235 

27 

34 

.26336 

.96332 

.23513 

.95349 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225 

26 

35 

.26864 

.96324 

.23541 

.95841 

.30209 

.95328 

.31863 

.94786 

.33518 

.94215 

25 

36 

.26392 

.96316 

.28569 

.95832 

.30237 

.95319 

.31S96 

.94777 

.33545 

.94206 

24 

37 

.26920 

.96308 

.23597 

.95824 

.30265 

.95310 

.31923 

.94768 

.33573 

.94196 

23 

33 

.26943 

.96301 

.28625 

.95316 

.30292 

.95301 

.31951 

.94758 

.33600 

.94186 

22 

39 

.26976 

.96293 

.23652 

.95807 

.30320 

.95293 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27004 

.96235 

.28680 

.95799 

.30348 

.95284 

.32006 

.94740 

.33655 

.94167 

20 

41 

.27032 

.96277 

.23703 

.95791 

.30376 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.27060 

.96269 

.28736 

.95782 

.30403 

.95266 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27033 

.96261 

.23764 

.95774 

.30431 

.95257 

.32089 

.94712 

.33737 

.94137 

17 

44 

.27116 

.96253 

.28792 

.95766 

.30459 

.95248 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28820 

.95757 

.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

.28847 

.95749 

.30514 

.95231 

.32171 

.94634 

.33819 

.94108 

14 

47 

.27200 

.96230 

.23375 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

48 

.27228 

.96222 

.23903 

.95732 

.30570 

.95213 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27256 

.96214 

.28931 

.95724 

.30597 

.95204 

32254 

.94656 

.33901 

.94078 

11 

50 

.27234 

.96206 

.23959 

.95715 

.30625 

.95195 

.32232 

.94646 

.33929 

.94068 

10 

51 

.27312 

.96193 

.23987 

.95707 

.30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

.30680 

.95177 

.32337 

.94627 

.33983 

.94049 

8 

53 

.27363 

.96182 

.29042 

.95690 

.30708 

.95163 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396 

.96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94609 

.34038 

.94029 

6 

55 

.27424 

.96166 

.29093 

.95673 

.30763 

.95150 

.32419 

.94599 

.34065 

.94019 

5 

56 

.27452 

.96153 

.29126 

.95664 

.30791 

.95142 

.32447 

.94590 

.34093 

.94009 

4 

57 

.27480 

.96150 

.29154 

.95656 

.30819 

.95133 

.32474 

.94580 

.34120 

.93999 

3 

58 

.27503 

.96142 

.29182 

.95647 

.30846 

.95124 

.32502 

.94571 

.34147 

.93989 

2 

59 

.27536 

.96134 

.29209 

.95639 

.30374 

.95115 

.32529 

.94561 

.34175 

.93979 

1 

60 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

.34202 

.93969 

0 

M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. | Cosin. 

Sine. 

Cosin. 

Sine. 

M. 


74-o 

73 'J 

723 

7IP 

703 








































224 


TABLE XIV. NATURAL SINES AND COSINES 



30° 

31° 

220 

230 

24° 


M. 

Sine. 

Cosin 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin 

M. 

0 

.34202 

.93969 

.35337 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91356 

60 

I 

.34229 

.93959 

.35864 

.93343 

.37488 

.92707 

.39100 

.92039 

.40700 

.91343 

59 

2 

.34257 

.93949 

.35891 

.93337 

.37515 

.92697 

.39127 

.92028 

.40727 

.91331 

58 

3 

.34234 

.93939 

.35918 

.93327 

.37542 

.92686 

.39153 

.92016 

.40753 

.91319 

57 

4 

34311 

.93929 

.35945 

.93316 

.37569 

.92675 

.391S0 

.92005 

.40780 

.91307 

56 

0 

.34339 

.93919 

.35973 

.93306 

.37595 

.92664 

.39207 

.91994 

.40306 

.91295 

55 

6 

.34366 

.93909 

.36000 

.93295 

.37622 

.92653 

.39234 

.91932 

.40333 

.91233 

54 

7 

.34393 

.93899 

.36027 

.93285 

.37649 

.92642 

.39260 

.91971 

.40860 

.91272 

53 

8 

.34421 

.93889 

.36054 

.93274 

.37676 

.92631 

.39237 

.91959 

.40886 

.91260 

52 

9 

.34448 

.93379 

.36031 

.93264 

.37703 

.92620 

.39314 

.91948 

.40913 

.91248 

51 

10 

.34475 

.93869 

.36108 

.93253 

.37730 

.92609 

.39341 

.91936 

.40939 

.91236 

50 

11 

.34503 

.93359 

.36135 

.93243 

.37757 

.92593 

.39367 

.91925 

.40966 

.91224 

49 

12 

.34530 

.93349 

.36162 

.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.93839 

.36190 

.93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200 

47 

14 

.34534 

.93329 

.36217 

.93211 

.37833 

.92565 

.39448 

.91891 

.41045 

.91188 

46 

15 

.34612 

.93319 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

45 

16 

.34639 

.93809 

.36271 

.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164 

44 

17 

.34666 

.93799 

.36298 

.93180 

.37919 

.92532 

.39528 

.91856 

.41125 

.91152 

43 

IS 

.34694 

.93789 

.36325 

.93169 

.37946 

.92321 

.39555 

.91845 

.41151 

.91140 

42 

19 

.34721 

.93779 

.36352 

.93159 

.37973 

.92510 

.39581 

.91833 

.41178 

.91128 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

.34775 

.93759 

.36406 

.93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104 

39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092 

38 

23 

.34330 

.93738 

.36461 

.93116 

.33080 

.92466 

.39688 

.91787 

.41234 

.91080 

37 

24 

.34357 

.93723 

.36438 

.93106 

.33107 

.92455 

.39715 

.91775 

.41310 

.91068 

36 

25 

.34334 

.93718 

.36515 

.93095 

.33134 

.92444 

.39741 

.91764 

.41337 

.91056 

35 

26 

.34912 

.93708 

.36542 

.93034 

.3316i 

.92432 

.39768 

.91752 

.41363 

.91044 

34 

27 

.34939 

.93693 

.36569 

.93074 

.33188 

.92421 

.39795 

.91741 

.41390 

.91032 

33 

23 

.34966 

.93638 

.36596 

.93063 

.38215 

.92410 

.39822 

.91729 

.41416 

.91020 

32 

29 

.34993 

.93677 

.36623 

.93052 

.38241 

.92399 

.39848 

.91718 

.41443 

.91008 

31 

30 

.35021 

.93667 

.36650 

.93042 

.33263 

.92338 

.39875 

.91706 

.41469 

.90996 

30 

31 

.35048 

.93657 

.36677 

.93031 

.33295 

.92377 

.39902 

.91694 

.41496 

.90984 

29 

32 

.35075 

.93647 

.36704 

.93020 

.33322 

.92366 

.39923 

.91633 

.41522 

.90972 

28 

33 

.35102 

.93637 

.36731 

.93010 

.33349 

.92355 

.39955 

.91671 

.41549 

.90960 

27 

34 

.35130 

.93626 

.36758 

.92999 

.33376 

.92343 

.39982 

.91660 

.41575 

.9094S 

26 

35 

.a5157 

.93616 

.36785 

.92983 

.38403 

.92332 

.40003 

.91648 

.41602 

.90936 

25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41623 

.90924 

24 

37 

.35211 

.93596 

.36339 

.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911 

23 

38 

.35239 

.93535 

.36367 

.92956 

.33433 

.92299 

.40088 

.91613 

.41681 

.90899 

22 

39 

.35266 

.93575 

.36394 

.92945 

.33510 

.92237 

.40115 

.91601 

.41707 

.90837 

21 

40 

.35293 

.93565 

.36921 

.92935 

.33537 

.92276 

.40141 

.91590 

.41734 

.90375 

20 

41 

.35.320 

.93555 

.36948 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760 

.90363 

19 

42 

.35347 

.93544 

.36975 

.92913 

.33591 

.92254 

.40195 

.91566 

.41787 

.90851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.33617 

.92243 

.40221 

.91555 

.41813 

.90839 

17 

44 

.35402 

.93524 

.37029 

.92392 

.33644 

.92231 

.40218 

.91543 

.41840 

.90S26 

16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.92220 

.40275 

.91531 

.41866 

.90314 

15 

46 

.35456 

.93503 

.37033 

.92370 

.33698 

.92209 

.40301 

.91519 

.41892 

.90802 

14 

47 

.35484 

.93493 

.37110 

.92359 

.33725 

.92198 

.40323 

.91508 

.41919 

.90790 

13 

48 

.35511 

.93483 

.37137 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.93472 

.37164 

.92838 

.33773 

.92175 

40381 

.91484 

A 1972 

.90766 

11 

50 

.35565 

.93462 

.37491 

.92827 

.38805 

.92164 

.40403 

.91472 

.41998 

.90753 

10 

51 

.35592 

.93452 

.37218 

.92316 

.38832 

.92152 

.40434 

.91461 

.42024 

.90741 

9 

52 

.35619 

.93441 

.37245 

.92805 

.38859 

.92141 

.40461 

.91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.33886 

.92130 

.4048S 

.91437 

.42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.92734 

.33912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.33939 

.92107 

.40541 

.91414 

.42130 

.90692 

5 

56 

.35728 

.93400 

.37353 

.92762 

.38966 

.92096 

.40567 

.91402 

.42156 

.90680 

4 

57 

.35755 

.93339 

.37330 

.92751 

.33993 

.92035 

.40594 

.91390 

.421 S3 

.90668 

3 

58 

.35732 

.93379 

.37407 

.92740 

.39020 

.92073 

.40621 

.91378 

.42209 

.90655 

2 

59 

.35810 

.93368 

.37434 

.92729 

.39046 

.92062 

.40617 

.91366 

.42235 

.90643 

1 

60 

.35337 

.93358 

.37461 

.92718 

.38073 

.92050 

.40674 

.91355 

.42262 

.90631 

0 

M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

M. 


690 

68° 

G7° 

66° 

65° 



















































TABLE XIV. NATURAL SINES AND COSINES 


225 





35° 

26° 

273 

28° 

1 39° 


M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

M. 

0 

.42262 

.90631 

.43837 

89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

60 

1 

.42288 

.90618 

.43363 

.89367 

.45425 

.89087 

.46973 

.88281 

.4S506 

.87448 

59 

2 

.42315 

..90306 

.43389 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

58 

3 

.42341 

.90594 

.43916 

.89341 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43963 

.89316 

.45529 

.89035 

.47076 

.88226 

.4S608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.8S213 

.48634 

.87377 

54 

7 

42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87362 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44093 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90433 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90453 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45737 

.88902 

.47332 

'.88089 

.48862 

.87250 

45 

16 

.42633 

.90433 

.44255 

.89674 

.45313 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48933 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42733 

.90333 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90353 

.44411 

.89597 

.45968 

.88808 

.47511 

.87993 

.49040 

.87150 

33 

23 

.42367 

90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42394 

.90334 

.44464 

.89571 

.46020 

.8S782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

23 

.42999 

.90284 

.44568 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.90271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.886S8 

.47741 

.87868 

.4926S 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44693 

.89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89423 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

33 

.43261 

.90158 

.44823 

.89339 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43237 

.90146 

,44854 

.89376 

.46407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44934 

.89311 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.4S099 

.87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

43 

.43523 

.90032 

.45088 

.89259 

.46639 

.88453 

.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89931 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89963 

.45218 

.89193 

.46767 

.88390 

.4S303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46319 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89913 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

53 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88303 

.45456 

.87476 

.49975 

.86617 

1 

60 

.43,337 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

0 

M. 

Cosin. 

Sinei 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

M. 


64° 

63° 

62° ‘ 

61° 

CO° 



11 





















































226 TABLE XIV. NATURAL SINES AND COSINES. 



303 

31° 

333 

I 330 

3410 


M. 

Sine. 

Cosin. 

Sine. 

Co Ad. 

Sine. 

Ccsin. 

Sine. 

Cosin. 

Sine. 

Cosin 

M. 

0 

.50000 

.86603 

51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.o2904 

60 

1 

.50025 

.86588 

51529 

.85702 

.53017 

.84789 

.544o8 

.83851 

.55943 

.82SS7 

59 

2 

.50050 

.86573 

51554 

.85687 

.53041 

.84774 

.54513 

.83335 

.55968 

.82871 

58 

3 

.50076- 

.86559 

.51579 

.85(72 

.53066 

.84759 

.54537 

.83819 

.55992 

.S2855 

57 


50101 

.86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.82822 

55 

6 

.50151 

.86515 

.51653 

.85627 

.53140 

.84712 

.54610 

.83772 

.56064 

.82806 

54 

ry 

l 

.50176 

.86501 

.51678 

.85612 

.53164 

.84697 

.54635 

.83756 

.56088 

.82790 

53 

8 

.50201 

.861S6 

.51703 

.35597 

.53189 

.84681 

.54659 

.83740 

.56112 

.82773 

52 

9 

.50227 

.88471 

.51728 

.85582 

.53214 

.84666 

.54683 

.83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.35567 

.53238 

.84650 

.54708 

.83708 

.56160 

.82741 

56 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708 

48 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.81604 

.54781 

.83660 

.56232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84.588 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.56280 

.82659 

45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83613 

.56305 

.82643 

44 

17 

.50423 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610 

42 

19 

.50478 

.86325 

.51977 

.85431 

.53460 

.84511 

.54927 

.83565 

.56377 

.82593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.81480 

.54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

.86281 

.52051 

.85385 

.53534 

.84164 

.54999 

.83517 

.56449 

.82544 

38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448 

.55024 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

.83435 

.56497 

.82511 

36 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495 

35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.56545 

.82473 

34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462 

33 

23 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84308 

.55242 

.83356 

.56689 

.82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55266 

.83340 

.56713 

.82363 

27 

31 

.50854 

.86101 

.52349 

.85203 

.53828 

.84277 

.55291 

.83324 

.56736 

.82347 

26 

35 

.50379 

.86089 

.52371 

.85188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.55339 

.83292 

.56784 

.82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55363 

.83276 

.56808 

.S2297 

23 

38 

.50954 

.86045 

.52443 

.85142 

.53926 

.84214 

.55338 

.83260 

.56832 

.82281 

22 

39 

.50979 

.86030 

.52473 

.85127 

.53951 

.84198 

.55412 

.83244 

.56356 

.82264 

21 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.52522 

.85096 

.54000 

.81167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.82214 

18 

43 

.51079 

.85970 

.52-572 

.85066 

.54049 

.84135 

.55509 

.83179 

.56952 

.82198 

17 

44 

.51104 

.35956 

.52597 

.85051 

.54073 

.84120 

.55533 

.83163 

.56976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.57000 

.82165 

15 

46 

.51154 

.85926 

.52616 

.85020 

.54122 

.84088 

.55581 

.83131 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.57047 

.82132 

13 

43 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

.57071 

.82115 

12 

49 

.51229 

.85331 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.57095 

.S2098 

11 

50 

.51254 

.85886 

.52745 

.84959 

.54220 

.84025 

.55678 

.83066 

.57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54269 

.83994 

.55726 

.83031 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017 

.57191 

.82032 

7 

54 

.51354 

.85806 

.52344 

.84897 

.54317 

.83962 

.55775 

.83001 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52393 

.84366 

.54366 

.83930 

.55823 

.82969 

.57262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.82953 

.57286 

.81965 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

2 

59 

1^51479 

.85732 

.52967 

.84S20 

.54440 

.83883 

.55895 

.82920 

.57334 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

.83367 

.55919 

.82904 

.57353 

.81915 

0 

M. 

CosinTl 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

A. 

— 

5.93 

583 

$70 

500 

550 

-J 




















































TABLE XIV. NATURAL SINES AND COSINES. 227 



350 

30° 

37 o 

38° 

39° 


M. 

Sine. 

Cosin 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin 

M. 

0 

.57358 

.81915 

.58779 

.80902 

.60182 

.79S64 

.61566 

.78S01 

.62932 

.77711 

60 

1 

.57381 

.81899 

.58802 

.80885 

.60205 

.79S46 

.61589 

.78783 

.62955 

.77694 

59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678 

58 

3 

.57429 

.81865 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77661 

57 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61958 

.78729 

.63022 

.77641 

56 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.7762£ 

55 

6 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694 

.63068 

.77605 

54 

7 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

.77581 

53 

8 

.57548 

.81782 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.7755C 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

49 

12 

.57643 

.81714 

.59061 

.80696 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.61864 

.78568 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80662 

.60506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

.63271 

.77439 

45 

16 

.57738 

.81647 

.59154 

.80627 

.60553 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610 

.60576 

.79565 

.61955 

.78496 

.63316 

.77402 

43 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338 

.77384 

42 

19 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460 

.63361 

.77366 

41 

20 

.57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.S0541 

.60668 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310 

38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78387 

.63451 

.77292 

37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.62115 

.78369 

.63473 

.77273 

36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183 

.78315 

.63540 

.77218 

33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

29 

.58047 

.81428 

.59459 

.80403 

.60853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

30 

.53070 

.81412 

.59482 

.80386 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

.58118 

.81378 

.59529 

.80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

28 

33 

.58141 

.81361 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.6096S 

.79264 

.62342 

.78188 

.63698 

.77088 

26 

35 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

.58236 

.81293 

.59646 

.80264 

.61038 

.79211 

.62411 

.78134 

.63765 

.77033 

23 

33 

.58260 

.81276 

.59669 

.80247 

.61061 

.79193 

.62433 

.78116 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.61084 

.79176 

.62456 

.78098 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977 

20 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 

.63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921 

17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79087 

.62570 

.78007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

.58449 

.81140 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966 

.76866 

14 

47 

.58472 

.81123 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79016 

.62660 

.77934 

.64011 

.76828 

12 

49 

.58519 

.81089 

.59926 

.80056 

.61314 

.78998 

62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.81033 

.59995 

.80003 

.61383 

.78944 

.62751 

.77861 

.64100 

.76754 

8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843 

.64123 

.76735 

7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

76717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

76698 

5 

56 

.58634 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.64190 

76679 

4 

57 

.58708 

.80953 

.60112 

.79916 

.61497 

.78855 

.62864 

.77769 

.64212 

76661 

3 

58 

.58731 

.80936 

.60135 

.79899 

.61520 

.78837 

.62887 

.77751 

.64234 

76642 

2 

59 

.58755 

.80919 

.60158 

.798S1 

.61543 

.78819 

.62909 

.77733 

.64256 

76623 

1 

60 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279 

76604 

0 

M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Gosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. ] 

tf. 


540 

530 

53° 1 

*51° 

50° 





















































228 TABLE XIV. NATURAL SINES AND COSINES 


[ 

400 

410 

420 

43 ° 

440 


M. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

M. 

0 

.64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.63200 

.73135 

.69166 

.71934 

60 

l 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.63221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.64346 

.76548 

.65672 

.75414 

.66973 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833 

55 

6 

.64412 

.76492 

.65733 

.75356 

.67043 

.74193 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.63349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.67036 

.74159 

.68370 

.72976 

.69633 

.71*72 

52 

9 

.64479 

.76436 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.65S25 

.75230 

.67129 

.74120 

.63412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76393 

.65347 

.75261 

.67151 

.74100 

.63434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76330 

.65869 

.75241 

.67172 

.74080 

.63455 

.72897 

.69717 

.71691 

48 

13 

.61568 

.76361 

.65891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

.64635 

.76304 

.65956 

.75165 

.67253 

.74002 

.68539 

.72817 

.69300 

.71610 

44 ' 

17 

.64657 

.76236 

.65978 

.75146 

.67280 

.73983 

.68561 

.72797 

.69321 

.71590 

43 

18 

.6-1679 

.76267 

.66000 

.75126 

.67301 

.73963 

. 685S2 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76243 

.66022 

.75107 

.67323 

.73944 

.63603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229 

.66044 

.75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

2! 

.64746 

.76210 

.66066 

.75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64763 

.76192 

.66083 

.75050 

.67387 

.73385 

.68666 

.72697 

.69925 

.7148S 

33 

23 

.64790 

.76173 

.66109 

.75030 

.67409 

.73365 

.63688 

.72677 

.69946 

.71468 

37 

24 

.64812 

.76154 

.66131 

.75011 

.67430 

.73346 

.63709 

.72657 

.69966 

.71447 

36 

25 

.64834 

.76135 

.66153 

.74992 

.67452 

.73826 

.63730 

.72637 

.69987 

.71427 

35 

26 

.64856 

.76116 

.66175 

.74973 

.67473 

.73806 

.68751 

.72617 

.70008 

.71407 

34 

27 

.64878 

.76097 

.66197 

.74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

28 

.64901 

.76078 

.66218 

.74934 

.67516 

•73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.64923 

.76059 

.66240 

.74915 

.67533 

.73747 

.68314 

.72557 

.70070 

.71345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.63835 

.72537 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66284 

.74876 

.67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

.76003 

.66306 

.74857 

.67602 

.73638 

.63878 

.72497 

.70132 

.71234 

28 

33 

.65011 

.75984 

.66327 

.74333 

.67623 

.73669 

.63399 

.72477 

.70153 

.71264 

27 

34 

.65033 

.75965 

.66349 

.74818 

.67645 

.73649 

.68920 

.72457 

.70174 

.71243 

26 

35 

.65055 

.75946 

.66371 

.74799 

.67666 

.73629 

.63941 

.72437 

.70195 

.71223 

25 

36 

.65077 

.75927 

.66393 

.74780 

.67688 

.73610 

.63962 

.72417 

.70215 

.71203 

24 

37 

.65100 

.75908 

.66414 

.74760 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

33 

.65122 

.75389 

.66436 

.74741 

.67730 

.73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144 

.75870 

.66453 

.74722 

.67752 

.73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75351 

.66430 

.74703 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65183 

.75332 

.66501 

.74683 

.67795 

.73511 

.69037 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.66523 

.74664 

.67816 

.73491 

.69033 

.72297 

.70339 

.71080 

18 

43 

.65232 

.75794 

.66545 

.74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

. 65254 

.75775 

.66566 

.74625 

.67859 

.73452 

.69130 

.72257 

.70331 

.71039 

16 

45 

.65276 

.75756 

.66588 

.74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65293 

.75733 

.66610 

.745S6 

.67901 

.73413 

.69172 

.72216 

.70422 

.70993 

14 

47 

.65320 

.75719 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65342 

.75700 

.66653 

.74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

.75680 

.66675 

.74523 

.67965 

.73353 

.69235 

.72156 

.70484 

.70937 

11 

50 

.65336 

.75661 

.66697 

.74509 

.67937 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

.74489 

.63008 

.73314 

.69277 

.72116 

.70525 

.70896 

9 

52 

.65430 

.75623 

.66740 

.74470 

.63029 

.73294 

.69298 

.72095 

.70546 

.70875 

8 

53 

.65452 

.75604 

.66762 

.74451 

.63051 

.73274 

.69319 

.72075 

.70567 

.70355 

7 

54 

.65474 

.75585 

.66783 

.74431 | 

.63072 

.73254 

.69340 

.72055 

.70587 

.70834 

6 

55 

.65496 

.75566 

.66305 

.74412 

.63093 

.73234 

.69361 

.72035 

.70608 

.70813 

5 

56 

.65518 

.75547 

.66327 

.74392 

.63115 

.73215 

.69382 

.72015 

.70628 

.70793 

4 

57 

.65540 

.75528 

.66348 

.74373 

.63136 

.73195 

.69403 

.71995 

.70649 

.70772 

3 

58 

.65562 

.75509 

.66370 

.74353 

.63157 

.73175 

.69424 

.71974 

.70670 

.70752 

2 

59 

.65584 

.75490 

.66391 

.74334 

.68179 

.73155 

.69445 

.71954 

.70690 

.70731 

1 

60 

.65606 

.75471 

.66913 

.74314 

.63200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 

M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine, i 

Cosin. 

Sine. 

Cosin. 

Sine. 

M. 


419^ 

483 

4-7° S 

4G° 

45° 






















































TABLE XV 


NATURAL TANGENTS AND COTANGENTS. 


230 TABLE XV. NATURAL TANGENTS AND COTANGENTS 



03 

lo 

2° 

30 

| 

M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363* 

.05241 

19.0811 

60 

] 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.93579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8321 

.03638 

27.4899 

.05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

,05416 

.18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708 

51 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

,05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.*6106 

45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

82 

.00640 

156.259 

02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.96S7 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.90S5 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02381 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20 8188 

.06554 

15 2571 

15 

46 

.01338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0S72 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

23.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


893 

88 ° 

870 

86 ° 




































TABLE XV. NATURAL TANGENTS AND COTANGENTS. 231 



40 

5° 

60 

7° 


M. 

Tang 

Cotang 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07030 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

6 

.07163 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9832 

.10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5093 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.S7895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

10952 

9.13093 

.12722 

7.86064 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.6S208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

33 

23 

.07812 

12.8014 

•09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.6124 

.09638 

10.3224 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

38 

.03104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

.13432 

7.44509 

21 

40 

.03163 

12.2505 

.09923 

10.0780 

.11638 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

8.53402 

.13491 

7.41240 

19 

42 

.03221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

11806 

8.47007 

.13580 

7.36389 

16 

45 

*.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

9.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11.5072 

.10452 

9.56791 

.12219 

8.18370 

13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11 4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


85° 

840 

830 

83° 


















































232 TABLE XV. NATURAL TANGENTS AND COTANGENTS, 



| 8° 

90 

10 ° 

lio 


M. 

I Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

60 , 

I 

.14084 

7.10038 

15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96S23 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.0S444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

-39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.73564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 ' 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.75S38 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

26. 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.86605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

25 

36 

.15124 

6.61219 

16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.S5013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20300 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

43 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

59 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

1 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

0 

| M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

VI. 

I 

810 

803 

790 

783 










































TABLE XV. NATURAL TANGENTS AND COTANGENTS. 233 


[ 

120 

133 

14t° 

150 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4; 00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24-995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30S60 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4 63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.5S001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

2t 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23354 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53563 

•23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3.605S8 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12S25 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.2S297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26603 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cctang. 

Tang. 

Cotang. 

Tang. 

M. 


770 

76 ° 

750 

7&o 






































234 TABLE XV. NATURAL TANGENTS AND COTANGENTS 



160 

1 70 

183 

190 

1 

M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.23895 

3.46030 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.8S511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.8S240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29034 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.348S9 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.31922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29363 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33293 

3.00319 

.35248 

2.83702 

35 

26 

.29195 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

• 3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 1 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98S68 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33636 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79S02 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30237 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

•34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09293 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


730 

720 

71° 

700 










































TABLE XV. NATURAL TANGENTS AND COTAin t GENTS 


235 



303 

210 


230 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

1 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46S38 

.42551 

2.35015 

57 

4 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.593S1 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.73017 

.33620 

2.58932 

.40640 

2.46065 

.42638 

2.34258 

53 

8 

.36661 

2.72771 

.33654 

2.5870S 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2 33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40309 

2.45043 

.42360 

2,33317 

48 

13 

.36326 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.33955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56487 

.41013 

2.43325 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2:32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41031 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41116 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.63892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.63414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37338 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

• .37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

39 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

.41S99 

2.38668 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41963 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38034 

.44071 

2.26909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.33020 

2.63021 

.40031 

2.49S07 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.33086 

2.62561 

.40098 

2.4 9386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.33120 

2.62332 

.40132 

2,49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44319 

2.25486 

5 

56 

.33253 

2.61418 

.40267 

2.48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.33353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

60 

.38336 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang: 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


69= 

683 

67 => 

66° 






































236 TABLE XV. NATURAL TANGENTS AND COTANGENTS 



24° 

25° 

200 

27° 

1 

M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.44523 

2.24604 

,46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

1 

.4455S 

2.24428 

.46666 

2.142S8 

.48309 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95S3S 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277 

53 

8 

.44S02 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 j 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.938S5 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49S94 

2.00423 

.52054 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45743 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

2.3 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45S89 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17532 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.9S828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.9S540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.4S306 

2.07014 

.50477 

1.98110 

.52687 

1.89801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50537 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.S9266 

9 

52 

.46348 

2.15760 

.4S486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.S9000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.8S734 

5 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50S79 

1.96544 

.53096 

1.SS337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

.53171 

1.SS073 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Co tang. 

Tang. 

Cotang. 

Tang. 

u. 

1 

N- 

65° 

64° 

630 I 

62° 
















































•AJBLE XV. NATURAL TANGENTS AND COTANGENTS. 237 



280 

390 

300 

310 


M 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.53171 

1.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

1.87941 

.55469 

1.80231 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

1.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

5S 

3 

.53283 

1.87677 

.55545 

1.80034 

.57851 

1.72357 

.60205 

1.66099 

57 

4 

.53320 

1.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

1.87415 

.55621 

1.79788 

.57929 

1.72625 

.60284 

1.65881 

55 

6 

.53395 

1.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

1.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

8 

.53470 

1.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

1.86391 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55812 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.6-4579 

43 

18 

.53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

1.85204 

.56270 

1.77713 

.5S59I 

1.70673 

.60960 

1.64041 

38 

23 

.54032 

1.85075 

.56309 

1.77592 

.58631 

1.-70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.56347 

1.77471 

.5S670 

1.70446 

.61040 

1.63826 

36 

25 

.54107 

1.84818 

.56335 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

1.846S9 

.53424 

1.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

1.84433 

.56501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

29 

.54258 

1.84305 

.56539 

1.76S69 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.56654 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

34 

.64446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83286 

.56346 

1.75913 

.59179 

1.6S979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1.75794 

.59218 

1.68866 

.61601 

1.62336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

.59258 

1.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.6S643 

.61681 

1.62125 

20 

41 

.54711 

1.82780 

.57000 

1.75437 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.57039 

1.75319 

.59376 

1.6S419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1.61808 

17 

44 

.54824 

1.82402 

.57116 

1.75082 

.59454 

1.68196 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

1.6S085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

.59533 

1.67974 

.61922 

1,61493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.61179 

11 

50 

.55051 

1.81649 

.57348 

1.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

.55089 

1.81524 

.57336 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67309 

62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

.59809 

1.67198 

.62204 

1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59S88 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

1.80901 

.57580 

1.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

1.66757 

.62366 

1.60345 

3 

58 

.55355 

1.80653 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.73321 

.60046 

1.66538 

.62446 

1.60137 

1 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 1 

Tang. 

Cotang. 

Tang. 

M. 


GIG 

603 

593 

580 

I 







































238 TABLE XV. NATURAL TANGENTS AND COTANGENTS 



320 

330 

340 

350 


M 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.62487 

1.60033 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

60 

I 

.62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70061 

1.42726 

59 

2 

.62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

5 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

70499 

1.41847 

49 

12 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.6S002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.68088 

1.46870 

.70673 

1.41497 

45 

16 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.70760 

1.41322 

43 

18 

.63217 

1.58184 

.65638 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58033 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51850 

.63366 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.6S429 

1.46137 

.71023 

1.40800 

37 

24 

.63462 

1.57575 

.65933 

1.51658 

.6S471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

34 

27 

.63584 

1.67271 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.6S6S5 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63371 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66140 

1.50512 

.6S985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56263 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64231 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.3S994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64401 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49 

.64437 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.3S314 

8 

53 

.64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72132 

1.38060 

5 

56 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

57 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

58 

.64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

59 

.64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

60 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


570 

5 

50 

550 

540 








































TABLE XV. NATURAL TANGENTS AND COTANGENTS. 239 



360 

370 

380 

390 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

1 31934 

.78551 

1.27306 

.81413 

1.22831 

51 

iO 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27i53 

.81510 

1.22685 

49 

12 

,73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

•1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

23 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

3.21454 

32 

29 

.73951 

1.35224 

.766S6 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

33 

.74131 

1.34896 

.76371 

1.30087 

.79686 

1.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

3.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

18 

43 

.74533 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74623 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

LI 9953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.23379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1,19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83S60 

1,19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


53° 

52° 

51° 

500 





































240 TABLE XV. NATURAL TANGENTS AND COTANGENTS 



4CP 

4:1° 

43° 

430 

m 

M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.33910 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87082 

1.14S34 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

1.06987 

56 

5 

.84153 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

6 

.8420-3 

1.18754 

.87236 

1.14632 

.90357 

1.10672 

.93578 

1.06362 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

8 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93638 

1.06738 

52 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

48 

13 

.84556 

1.18264 

.87595 

1.14162 

.90727 

1.10220 

.93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.18125 

.87693 

1.14028 

.90834 

1.10091 

.94074 

1.06303 

45 

16 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1.10027 

.94)25 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13394 

.90940 

1.09963 

.94180 

1.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993 

1.09399 

.94235 

1.06147 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046 

1.09834 

.94290 

1.06056 

41 

20 

.84906 

1.17777 

.87955 

. 1.13694 

.91099 

1.09770 

.94345 

1.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17633 

.88059 

1.13561 

.91205 

1.09642 

.94455 

1.05870 

38 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

35 

as 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09336 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13223 

.91473 

1.09322 

.94731 

1.05562 

33 

23 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

32 

29 

.85358 

1.17154 

.88421 

1.13096 

.91530 

1.09195 

.94841 

1.05439 

31 

30 

.85403 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

32 

.85509 

1.16947 

.88576 

1.12397 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88623 

1.12331 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1.16809 

.83680 

1.12765 

.91847 

1.03876 

.95118 

1.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.03813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955 

1.08749 

.95229 

1.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.03636 

.95284 

1.04949 

23 

38 

.85311 

1.16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

1.04888 

22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.08559 

.95395 

1.04827 

21 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

19 

42 

.86014 

1.16261 

.89097 

1.12233 

.92277 

1.08369 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

L07990 

.95897 

1.04279 

12 

49 

.86368 

1.15783 

.89463 

1.11778 

.92655 

1.07927 

.95952 

1.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

9 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07733 

.96120 

1.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1.07676 

.96176 

1.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92930 

1.07550 

.96288 

1.03855 

5 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

1.03794 

4 

57 

.86776 

1.15240 

.89383 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

58 

.86327 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

1.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

1 

60 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96569 

1.03553 

0 

M. 

[Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

EL 

- - 

1 493 

483 

470 

4:03 









































TABLE XV. NATURAL TANGENTS AND COTANGENTS. 241 



44 ° 



440 



440 


M. 

Tang. 

Cotang. 

M. 

M. 

Tang. 

Cotang. 

M. 

M. 

Tang. 

Cotang. 

M. 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

39 

41 

.9S901 

1.01112 

19 

2 

.96631 

1.03433 

58 

22 

.97813 

1.02236 

38 

42 

.93958 

1.01053 

18 

3 

.96733 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

36 

44 

.99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

35 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

26 

.93041 

1.01993 

34 

46 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.93098 

1.01939 

33 

47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

52 

23 

.93155 

1.01879 

32 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.93213 

1.01820 

31 

49 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

30 

.9S270 

1.01761 

30 

50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.93327 

1.01702 

29 

51 

.99478 

1.00525 

9 

12 

.97246 

1.02332 

48 

32 

.93334 

1.01642 

28 

52 

.99536 

1.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

1.01583 

27 

53 

.99594 

1.00408 

7 

14 

.97359 

1.02713 

46 

34 

.93499 

1.01524 

26 

54 

.99652 

1.00350 

6 

15 

.97413 

1.02653 

45 

35 

.98556 

1.01465 

25 

55 

.99710 

1.00291 

5 

16 

.974:2 

1.02593 

44 

36 

.93613 

1.01406 

24 

56 

.99763 

1.00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23 

57 

.99326 

1.00175 

O 

0 

18 

.97586 

1.02474 

42 

33 

.98728 

1.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

M. 

Cotang. 

Tang. 

M. 

M. 

Cotang. 

Tang. 

M. 

M. 

Cotang. 

Tang. 

M. 


450 



450 



453 
















































242 TABLE XVI. RISE PER MILE OF VARIOUS GRADES, 


TABLE XVI. 

RISE PER MILE OP VARIOUS GRADES. 


Grade 

per 

Station. 

Rise per 
Mile- 

Grade 

per 

Station. 

Rise per 
Mile. 

Grade 

per 

Station. 

Rise per 
Mile. 

Grade 

per 

Station. 

Rise per 
Mile. 

.01 

.523 

.41 

21.648 

.81 

42.768 

1.21 

63.888 

.02 

1.056 

.42 

22.176 

.82 

43.296 

1.22 

64.416 

.03 

1.584 

.43 

22.704 

.83 

43.824 

1.23 

64.944 

.04 

2.112 

.44 

23.232 

.84 

44.352 

1.24 

65.472 

.05 

2.640 

.45 

23.760 

.85 

44.880 

1.25 

66.000 

.06 

3.168 

.46 

24.233 

.86 

45.403 

1.26 

66.528 

.07 

3.696 

.47 

24.816 

.87 

45.936 

1.27 

67.056 

.08 

4.224 

.48 

25.344 

.83 

46.464 

1.28 

67.584 

.09 

4.752 

.49 

25.872 

.89 

46.992 

1.29 

68.112 

.10 

5.280 

.50 

26.400 

.90 

47.520 

1.30 

6S.640 

.11 

5.808 

.51 

26.923 

.91 

48.048 

1.31 

69.163 

.12 

6.336 

.52 

27.456 ' 

.92 

48.576 

1.32 

69.696 

.13 

6.864 

.53 

27.984 

.93 

49.104 

1.33 

70.224 

.14 

7.392 

.54 

28.512 

.94 

49.632 

1.34 

70.752 

.15 

7.920 

.55 

29.040 

.95 

50.160 

1.35 

71.280 

.16 

8.443 

:56 

29.568 

.96 

50.688 

1.36 

71.808 ' 

.17 

8.976 

.57 

30.096 

.97 

51.216 

1.37 

72.336 

.18 

9.504 

.53 

30.624 

.93 

51.744 

1.33 « 

» 72.864 

.19 

10.032 

.59 

31.152 

.99 

52.272 

1.39 

73.392 

.20 

10.560 

.60 

31.680 

1.00 

52.800 

1.40 

73.920 

.21 

11.088 

.61 

32.208 

l.oi 

53.323 

1.41 

74.448 

.22 

11.616 

.62 

32.736 

1.02 

53.856 

1.42 

74.976 

.23 

12.144 

.63 

33.264 

1.03 

54.384 

1.43 

75.504 

.24 

12.672 

.64 

33.792 

1.04 

54.912 

1.44 

76.032 

.25 

13.200 

.65 

34.320 

1.05 

55.440 

1.45 

76.560 

.26 

13.723 

.66 

34.848 - 

1.06 

55.963 

1.46 

77.088 

.27 

14.256 

.67 

35.376 

1.07 

56.496 

1.47 

77.616 

.23 

14.784 

.68 

35.904 

1.03 

57.024 

1.48 

78.144 

v .29 

15.312 

.69 

36.432 

1.09 

57.552 

1.49 

78.672 


15.840 

.70 

36.960 

1.10 

58.080 

1.50 

79.200 


16.363 

.71 

37.488 

1.11 

58.603 

1.51 

79.728 

•432 

16.896 

.72 

38.016 

1.12 

59.136 

1.52 

80.256 

.33 

17.424 

.73 

38.544 

1.13 

59.664 

1.53 

80.784 

.34 

17.952 

.74 

39.072 

1.14 

60.192 

1.54 

81.312 

.35 

18.430 

.75 

39.600 

1.15 

60.720 

1.55 

81.840 

.36 

19.003 

.76 

40.123 

1.16 

61.248 

1.56 

82.368 


- 19.536 

.77 

40.656 

1.17 

61.776 

1.57 

82.896 

.33®* 

?•-20.064 

.78 

41.184 

1.18 

62.304 

1.58 

83.424 

.39 

20.592 

.79 

41.712 

1.19 

62.832 

1.59 

83.952 

.40 

21.12Q 

.80 

42.240 

1.20 

63.360 

1.60 

84.480 



















TABLE XVI. RISE PER MILE OF VARIOUS GRADES. 243 


Grads 

per 

Station. 


1.61 

1.62 

1.63 

1.64 

1.65 

1.66 
1.67 
1.63 

1.69 

1.70 


1.71 

1.72 

1.73 

1.74 

1.75 

1.76 

1.77 
1.73 

1.79 

1.80 


Rise per 
Mile. 


85.003 

85.536 

86.064 

86.592 

87.120 

87.648 

88.176 

88.704 

89.232 

89.760 

90.288 

90.816 

91.344 

91.872 

92.400 

92.928 

93.456 

93.984 

94.512 

95.040 


Grade 

per 

Station. 


1.81 

1.82 

1.83 

1.84 

1.85 
1.S6 

1.87 

1.88 

1.89 

1.90 


1.91 

1.92 

1.93 

1.94 

1.95 

1.96 

1.97 

1.98 

1.99 
2.00 


Rise per 
Mile. 


95.563 

96.096 

96.624 

97.152 

97.630 

93.208 

93.736 

99.264 

99.792 

100.320 

100.848 

101.376 

101.904 

102.432 

102.960 

103.488 

104.016 

104.544 

105.072 

105.600 


Grade 

per 

Station. 

_ 


2.10 

2.20 

2.30 

2.40 
2.50 
2.60 

2.70 
2.80 

2.90 
3.00 

3.10 

3.20 

3.30 

3.40 
? 50 
3.60 

3.70 
3.80 

3.90 
4.00 


Rise per 
Mile. 


110.880 

116.160 

121.440 

126.720 

132.000 

137.280 

142.560 

147.840 

153.120 

158.400 

163.680 

163.960 

174.240 

179.520 

184.800 

190.080 

195.360 

200.640 

205.920 

211.200 


Grade 

per 

Station. 


4.10 

4.20 

4.30 

4.40 

4.50 

4.60 

4.70 

4.80 

4.90 

5.00 


5.10 

5.20 

5.30 

5.40 

5.50 

5.60 

5.70 

5.80 

5.90 

6.00 


Rise per 
Mile. 


216.480 

221.760 

227.040 

232.320 

237.600 

242.880 

248.160 

253.440 

258.720 

264.000 

269.280 

274.560 

279.840 

285.120 

290.400 

295.680 

300.960 

306.240 

311.520 

316.800 


niE END 


























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